TPTP Problem File: SLH0414^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Median_Method/0000_Median/prob_00347_013114__14824140_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1440 ( 485 unt; 167 typ; 0 def)
% Number of atoms : 4046 (1148 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 12007 ( 493 ~; 83 |; 294 &;9124 @)
% ( 0 <=>;2013 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 831 ( 831 >; 0 *; 0 +; 0 <<)
% Number of symbols : 164 ( 161 usr; 15 con; 0-5 aty)
% Number of variables : 3649 ( 190 ^;3330 !; 129 ?;3649 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:45:29.867
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (161)
thf(sy_c_Countable__Set_Ocountable_001t__Nat__Onat,type,
counta1168086296615599829le_nat: set_nat > $o ).
thf(sy_c_Countable__Set_Ocountable_001tf__a,type,
counta4098120917673242425able_a: set_a > $o ).
thf(sy_c_Countable__Set_Ofrom__nat__into_001t__Nat__Onat,type,
counta7321652538601044515to_nat: set_nat > nat > nat ).
thf(sy_c_Countable__Set_Ofrom__nat__into_001tf__a,type,
counta1652060073399151467into_a: set_a > nat > a ).
thf(sy_c_Disjoint__Sets_Opartition__on_001t__Nat__Onat,type,
disjoi4774308525696689793on_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
inj_on_nat_a: ( nat > a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Nat__Onat,type,
inj_on_a_nat: ( a > nat ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
the_inv_into_nat_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001tf__a,type,
the_inv_into_nat_a: set_nat > ( nat > a ) > a > nat ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Nat__Onat,type,
the_inv_into_a_nat: set_a > ( a > nat ) > nat > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__a,type,
the_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__Nat__Onat_J,type,
semila1667268886620078168et_nat: ( set_nat > set_nat > set_nat ) > set_nat > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__order_001t__Nat__Onat,type,
semila1248733672344298208er_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__order_001t__Set__Oset_It__Nat__Onat_J,type,
semila2291775939624898198et_nat: ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001tf__a,type,
lattic1148846883994911187_nat_a: ( nat > a ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001tf__a,type,
lattic3288624042836100505on_a_a: ( a > a ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3014633134055518761et_nat: set_set_nat > set_nat ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Nat__Onat,type,
lattic6009151579333465974et_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3109210760196336428et_nat: ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001tf__a,type,
lattic5078705180708912344_set_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001t__Nat__Onat,type,
lattic1029310888574255042et_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001tf__a,type,
lattic5961991414251573132_set_a: ( a > a > a ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Nat__Onat,type,
lattic7742739596368939638_F_nat: ( nat > nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001tf__a,type,
lattic5116578512385870296ce_F_a: ( a > a > a ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3835124923745554447et_nat: set_set_nat > set_nat ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Nat__Onat,type,
measur1302623347068717141at_nat: set_set_nat > ( set_nat > nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
measur5248428813077667851et_nat: set_set_nat > ( set_nat > set_nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001tf__a,type,
measur2960000890325386681_nat_a: set_set_nat > ( set_nat > a ) > $o ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Nat__Onat_001t__Nat__Onat,type,
measur4601247141005857854at_nat: nat > nat > ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
measur6959333727155881972et_nat: nat > nat > ( nat > set_nat ) > nat > nat > nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Nat__Onat_001tf__a,type,
measur1338148430225957392_nat_a: nat > nat > ( nat > a ) > nat > nat > nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
measur3393698822500512756at_nat: set_nat > set_nat > ( set_nat > nat ) > set_nat > set_nat > set_nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
measur5257060982548205482et_nat: set_nat > set_nat > ( set_nat > set_nat ) > set_nat > set_nat > set_nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_It__Nat__Onat_J_001tf__a,type,
measur7612622215237796826_nat_a: set_nat > set_nat > ( set_nat > a ) > set_nat > set_nat > set_nat ).
thf(sy_c_Measure__Space_Osup__lexord_001tf__a_001t__Nat__Onat,type,
measur6529588965902446770_a_nat: a > a > ( a > nat ) > a > a > a ).
thf(sy_c_Measure__Space_Osup__lexord_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
measur2251796063117055080et_nat: a > a > ( a > set_nat ) > a > a > a ).
thf(sy_c_Measure__Space_Osup__lexord_001tf__a_001tf__a,type,
measur6235662355876231836rd_a_a: a > a > ( a > a ) > a > a > a ).
thf(sy_c_Median_Odown__ray_001t__Nat__Onat,type,
down_ray_nat: set_nat > $o ).
thf(sy_c_Median_Odown__ray_001tf__a,type,
down_ray_a: set_a > $o ).
thf(sy_c_Median_Ointerval_001t__Nat__Onat,type,
interval_nat: set_nat > $o ).
thf(sy_c_Median_Ointerval_001tf__a,type,
interval_a: set_a > $o ).
thf(sy_c_Median_Oup__ray_001t__Nat__Onat,type,
up_ray_nat: set_nat > $o ).
thf(sy_c_Median_Oup__ray_001tf__a,type,
up_ray_a: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_It__Nat__Onat_J,type,
ord_Least_set_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001tf__a,type,
ord_Least_a: ( a > $o ) > a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mtf__a_J,type,
ord_less_eq_o_a: ( $o > a ) > ( $o > a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
order_Greatest_a: ( a > $o ) > a ).
thf(sy_c_Orderings_Oordering_001t__Nat__Onat,type,
ordering_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001tf__a,type,
ordering_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Nat__Onat,type,
partia6822818058636336922ng_nat: ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Set__Oset_It__Nat__Onat_J,type,
partia5623167761149600464et_nat: ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001tf__a,type,
partia125584492769400372ring_a: ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Nat__Onat,type,
preordering_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Set__Oset_It__Nat__Onat_J,type,
preordering_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001tf__a,type,
preordering_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__a,type,
vimage_nat_a: ( nat > a ) > set_a > set_nat ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Nat__Onat,type,
vimage_a_nat: ( a > nat ) > set_nat > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__a,type,
vimage_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001tf__a,type,
set_or672772299803893939Most_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001tf__a,type,
set_or5139330845457685135Than_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Nat__Onat_J,type,
set_or1731685050470061051et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001tf__a,type,
set_ord_atLeast_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001tf__a,type,
set_ord_atMost_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or7074010630789208630et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001tf__a,type,
set_or4472690218693186638Most_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or8625682525731655386et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001tf__a,type,
set_or5939364468397584554Than_a: a > a > set_a ).
thf(sy_c_Sigma__Algebra_Oalgebra_001t__Nat__Onat,type,
sigma_algebra_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Obinary_001t__Nat__Onat,type,
sigma_binary_nat: nat > nat > nat > nat ).
thf(sy_c_Sigma__Algebra_Obinaryset_001t__Nat__Onat,type,
sigma_binaryset_nat: set_nat > set_nat > nat > set_nat ).
thf(sy_c_Sigma__Algebra_Oclosed__cdi_001t__Nat__Onat,type,
sigma_closed_cdi_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001t__Nat__Onat,type,
sigma_8817008012692346403ra_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001tf__a,type,
sigma_4968961713055010667ebra_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_b,type,
b: a ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1272)
thf(fact_0_assms_I1_J,axiom,
interval_a @ i ).
% assms(1)
thf(fact_1_assms_I5_J,axiom,
member_a @ b @ i ).
% assms(5)
thf(fact_2_assms_I4_J,axiom,
member_a @ a2 @ i ).
% assms(4)
thf(fact_3_assms_I3_J,axiom,
ord_less_eq_a @ x @ b ).
% assms(3)
thf(fact_4_assms_I2_J,axiom,
ord_less_eq_a @ a2 @ x ).
% assms(2)
thf(fact_5_interval__def,axiom,
( interval_a
= ( ^ [I: set_a] :
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ I )
=> ( ( member_a @ Z @ I )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z )
=> ( member_a @ Y @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_6_interval__def,axiom,
( interval_nat
= ( ^ [I: set_nat] :
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ I )
=> ( ( member_nat @ Z @ I )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( member_nat @ Y @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_7_order__refl,axiom,
! [X2: a] : ( ord_less_eq_a @ X2 @ X2 ) ).
% order_refl
thf(fact_8_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_9_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_10_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_11_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_12_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_13_up__ray__def,axiom,
( up_ray_a
= ( ^ [I: set_a] :
! [X: a,Y: a] :
( ( member_a @ X @ I )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( member_a @ Y @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_14_up__ray__def,axiom,
( up_ray_nat
= ( ^ [I: set_nat] :
! [X: nat,Y: nat] :
( ( member_nat @ X @ I )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( member_nat @ Y @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_15_down__ray__def,axiom,
( down_ray_a
= ( ^ [I: set_a] :
! [X: a,Y: a] :
( ( member_a @ Y @ I )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( member_a @ X @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_16_down__ray__def,axiom,
( down_ray_nat
= ( ^ [I: set_nat] :
! [X: nat,Y: nat] :
( ( member_nat @ Y @ I )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( member_nat @ X @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_17_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_18_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_19_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_20_nle__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_eq_a @ A @ B ) )
= ( ( ord_less_eq_a @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_21_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_22_le__cases3,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( ( ord_less_eq_a @ X2 @ Y2 )
=> ~ ( ord_less_eq_a @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_a @ Y2 @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_a @ X2 @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_a @ Z2 @ Y2 )
=> ~ ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_a @ Z2 @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_23_le__cases3,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_24_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ( ord_less_eq_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_25_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_26_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_27_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_28_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_29_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_30_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_31_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_32_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_33_order__antisym,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_34_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_35_order__antisym,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_36_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_37_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_38_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_39_order__antisym__conv,axiom,
! [Y2: a,X2: a] :
( ( ord_less_eq_a @ Y2 @ X2 )
=> ( ( ord_less_eq_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_40_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_41_order__antisym__conv,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_42_linorder__le__cases,axiom,
! [X2: a,Y2: a] :
( ~ ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_43_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_44_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_45_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_46_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_47_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_48_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_49_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_50_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > a,C: a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_51_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_52_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_53_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_54_ord__eq__le__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_55_ord__eq__le__subst,axiom,
! [A: set_nat,F: a > set_nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_56_ord__eq__le__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_57_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_58_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_59_ord__eq__le__subst,axiom,
! [A: a,F: set_nat > a,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_60_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_61_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_62_linorder__linear,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
| ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_63_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_64_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_65_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_66_order__eq__refl,axiom,
! [X2: a,Y2: a] :
( ( X2 = Y2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_67_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_68_order__eq__refl,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_69_order__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_70_order__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_71_order__subst2,axiom,
! [A: a,B: a,F: a > set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_72_order__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_73_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_74_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_75_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > a,C: a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_76_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_77_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_78_order__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_79_order__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_80_order__subst1,axiom,
! [A: a,F: set_nat > a,B: set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_81_order__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_82_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_83_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_84_order__subst1,axiom,
! [A: set_nat,F: a > set_nat,B: a,C: a] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_85_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_86_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_87_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ( ord_less_eq_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_88_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_89_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_90_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_91_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_92_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_93_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_94_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_95_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_96_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_97_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_98_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_99_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ( ord_less_eq_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_100_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_101_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_102_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_103_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_104_order__trans,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ( ord_less_eq_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_105_order__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_106_order__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_107_Greatest__equality,axiom,
! [P: a > $o,X2: a] :
( ( P @ X2 )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X2 ) )
=> ( ( order_Greatest_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_108_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_109_Greatest__equality,axiom,
! [P: set_nat > $o,X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y4: set_nat] :
( ( P @ Y4 )
=> ( ord_less_eq_set_nat @ Y4 @ X2 ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_110_GreatestI2__order,axiom,
! [P: a > $o,X2: a,Q: a > $o] :
( ( P @ X2 )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X2 ) )
=> ( ! [X3: a] :
( ( P @ X3 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_111_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_112_GreatestI2__order,axiom,
! [P: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y4: set_nat] :
( ( P @ Y4 )
=> ( ord_less_eq_set_nat @ Y4 @ X2 ) )
=> ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_113_le__left__mono,axiom,
! [X2: a,Y2: a,A: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ A )
=> ( ord_less_eq_a @ X2 @ A ) ) ) ).
% le_left_mono
thf(fact_114_le__left__mono,axiom,
! [X2: nat,Y2: nat,A: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ A )
=> ( ord_less_eq_nat @ X2 @ A ) ) ) ).
% le_left_mono
thf(fact_115_le__left__mono,axiom,
! [X2: set_nat,Y2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ A ) ) ) ).
% le_left_mono
thf(fact_116_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_a
= ( ^ [X4: $o > a,Y6: $o > a] :
( ( ord_less_eq_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_117_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X4: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_118_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X4: $o > set_nat,Y6: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_119_order_Opartial__preordering__axioms,axiom,
partia125584492769400372ring_a @ ord_less_eq_a ).
% order.partial_preordering_axioms
thf(fact_120_order_Opartial__preordering__axioms,axiom,
partia6822818058636336922ng_nat @ ord_less_eq_nat ).
% order.partial_preordering_axioms
thf(fact_121_order_Opartial__preordering__axioms,axiom,
partia5623167761149600464et_nat @ ord_less_eq_set_nat ).
% order.partial_preordering_axioms
thf(fact_122_increasing__def,axiom,
( measur2960000890325386681_nat_a
= ( ^ [M: set_set_nat,Mu: set_nat > a] :
! [X: set_nat] :
( ( member_set_nat @ X @ M )
=> ! [Y: set_nat] :
( ( member_set_nat @ Y @ M )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_a @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_123_increasing__def,axiom,
( measur1302623347068717141at_nat
= ( ^ [M: set_set_nat,Mu: set_nat > nat] :
! [X: set_nat] :
( ( member_set_nat @ X @ M )
=> ! [Y: set_nat] :
( ( member_set_nat @ Y @ M )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_124_increasing__def,axiom,
( measur5248428813077667851et_nat
= ( ^ [M: set_set_nat,Mu: set_nat > set_nat] :
! [X: set_nat] :
( ( member_set_nat @ X @ M )
=> ! [Y: set_nat] :
( ( member_set_nat @ Y @ M )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_125_increasingD,axiom,
! [M2: set_set_nat,F: set_nat > a,X2: set_nat,Y2: set_nat] :
( ( measur2960000890325386681_nat_a @ M2 @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( member_set_nat @ X2 @ M2 )
=> ( ( member_set_nat @ Y2 @ M2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_126_increasingD,axiom,
! [M2: set_set_nat,F: set_nat > nat,X2: set_nat,Y2: set_nat] :
( ( measur1302623347068717141at_nat @ M2 @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( member_set_nat @ X2 @ M2 )
=> ( ( member_set_nat @ Y2 @ M2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_127_increasingD,axiom,
! [M2: set_set_nat,F: set_nat > set_nat,X2: set_nat,Y2: set_nat] :
( ( measur5248428813077667851et_nat @ M2 @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( member_set_nat @ X2 @ M2 )
=> ( ( member_set_nat @ Y2 @ M2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_128_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_129_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_132_LeastI2__wellorder__ex,axiom,
! [P: nat > $o,Q: nat > $o] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [A3: nat] :
( ( P @ A3 )
=> ( ! [B4: nat] :
( ( P @ B4 )
=> ( ord_less_eq_nat @ A3 @ B4 ) )
=> ( Q @ A3 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_133_LeastI2__wellorder,axiom,
! [P: nat > $o,A: nat,Q: nat > $o] :
( ( P @ A )
=> ( ! [A3: nat] :
( ( P @ A3 )
=> ( ! [B4: nat] :
( ( P @ B4 )
=> ( ord_less_eq_nat @ A3 @ B4 ) )
=> ( Q @ A3 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_wellorder
thf(fact_134_Least__equality,axiom,
! [P: a > $o,X2: a] :
( ( P @ X2 )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ X2 @ Y4 ) )
=> ( ( ord_Least_a @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_135_Least__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ X2 @ Y4 ) )
=> ( ( ord_Least_nat @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_136_Least__equality,axiom,
! [P: set_nat > $o,X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y4: set_nat] :
( ( P @ Y4 )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) )
=> ( ( ord_Least_set_nat @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_137_LeastI2__order,axiom,
! [P: a > $o,X2: a,Q: a > $o] :
( ( P @ X2 )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ X2 @ Y4 ) )
=> ( ! [X3: a] :
( ( P @ X3 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ X3 @ Y5 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_138_LeastI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ X2 @ Y4 ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ X3 @ Y5 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_139_LeastI2__order,axiom,
! [P: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y4: set_nat] :
( ( P @ Y4 )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) )
=> ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ X3 @ Y5 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least_set_nat @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_140_Least1__le,axiom,
! [P: a > $o,Z2: a] :
( ? [X5: a] :
( ( P @ X5 )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ X5 @ Y4 ) )
& ! [Y4: a] :
( ( ( P @ Y4 )
& ! [Ya: a] :
( ( P @ Ya )
=> ( ord_less_eq_a @ Y4 @ Ya ) ) )
=> ( Y4 = X5 ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq_a @ ( ord_Least_a @ P ) @ Z2 ) ) ) ).
% Least1_le
thf(fact_141_Least1__le,axiom,
! [P: nat > $o,Z2: nat] :
( ? [X5: nat] :
( ( P @ X5 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ X5 @ Y4 ) )
& ! [Y4: nat] :
( ( ( P @ Y4 )
& ! [Ya: nat] :
( ( P @ Ya )
=> ( ord_less_eq_nat @ Y4 @ Ya ) ) )
=> ( Y4 = X5 ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ Z2 ) ) ) ).
% Least1_le
thf(fact_142_Least1__le,axiom,
! [P: set_nat > $o,Z2: set_nat] :
( ? [X5: set_nat] :
( ( P @ X5 )
& ! [Y4: set_nat] :
( ( P @ Y4 )
=> ( ord_less_eq_set_nat @ X5 @ Y4 ) )
& ! [Y4: set_nat] :
( ( ( P @ Y4 )
& ! [Ya: set_nat] :
( ( P @ Ya )
=> ( ord_less_eq_set_nat @ Y4 @ Ya ) ) )
=> ( Y4 = X5 ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq_set_nat @ ( ord_Least_set_nat @ P ) @ Z2 ) ) ) ).
% Least1_le
thf(fact_143_Least1I,axiom,
! [P: a > $o] :
( ? [X5: a] :
( ( P @ X5 )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ X5 @ Y4 ) )
& ! [Y4: a] :
( ( ( P @ Y4 )
& ! [Ya: a] :
( ( P @ Ya )
=> ( ord_less_eq_a @ Y4 @ Ya ) ) )
=> ( Y4 = X5 ) ) )
=> ( P @ ( ord_Least_a @ P ) ) ) ).
% Least1I
thf(fact_144_Least1I,axiom,
! [P: nat > $o] :
( ? [X5: nat] :
( ( P @ X5 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ X5 @ Y4 ) )
& ! [Y4: nat] :
( ( ( P @ Y4 )
& ! [Ya: nat] :
( ( P @ Ya )
=> ( ord_less_eq_nat @ Y4 @ Ya ) ) )
=> ( Y4 = X5 ) ) )
=> ( P @ ( ord_Least_nat @ P ) ) ) ).
% Least1I
thf(fact_145_Least1I,axiom,
! [P: set_nat > $o] :
( ? [X5: set_nat] :
( ( P @ X5 )
& ! [Y4: set_nat] :
( ( P @ Y4 )
=> ( ord_less_eq_set_nat @ X5 @ Y4 ) )
& ! [Y4: set_nat] :
( ( ( P @ Y4 )
& ! [Ya: set_nat] :
( ( P @ Ya )
=> ( ord_less_eq_set_nat @ Y4 @ Ya ) ) )
=> ( Y4 = X5 ) ) )
=> ( P @ ( ord_Least_set_nat @ P ) ) ) ).
% Least1I
thf(fact_146_subsetI,axiom,
! [A4: set_a,B5: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B5 ) )
=> ( ord_less_eq_set_a @ A4 @ B5 ) ) ).
% subsetI
thf(fact_147_subsetI,axiom,
! [A4: set_nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B5 ) )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_148_subset__antisym,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_149_in__mono,axiom,
! [A4: set_a,B5: set_a,X2: a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( member_a @ X2 @ A4 )
=> ( member_a @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_150_in__mono,axiom,
! [A4: set_nat,B5: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ X2 @ A4 )
=> ( member_nat @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_151_subsetD,axiom,
! [A4: set_a,B5: set_a,C: a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_152_subsetD,axiom,
! [A4: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_153_equalityE,axiom,
! [A4: set_nat,B5: set_nat] :
( ( A4 = B5 )
=> ~ ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ~ ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_154_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B6: set_a] :
! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_a @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_155_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_156_equalityD1,axiom,
! [A4: set_nat,B5: set_nat] :
( ( A4 = B5 )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_157_equalityD2,axiom,
! [A4: set_nat,B5: set_nat] :
( ( A4 = B5 )
=> ( ord_less_eq_set_nat @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_158_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B6: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_159_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_160_subset__refl,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ A4 ) ).
% subset_refl
thf(fact_161_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_162_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_163_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_164_subset__trans,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_165_empty__subsetI,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).
% empty_subsetI
thf(fact_166_subset__empty,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_167_atLeast__subset__iff,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ X2 ) @ ( set_ord_atLeast_a @ Y2 ) )
= ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_168_atLeast__subset__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or1731685050470061051et_nat @ X2 ) @ ( set_or1731685050470061051et_nat @ Y2 ) )
= ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_169_atLeast__subset__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ X2 ) @ ( set_ord_atLeast_nat @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_170_atMost__subset__iff,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ X2 ) @ ( set_ord_atMost_a @ Y2 ) )
= ( ord_less_eq_a @ X2 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_171_atMost__subset__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X2 ) @ ( set_or4236626031148496127et_nat @ Y2 ) )
= ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_172_atMost__subset__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_173_Int__subset__iff,axiom,
! [C2: set_nat,A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( ( ord_less_eq_set_nat @ C2 @ A4 )
& ( ord_less_eq_set_nat @ C2 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_174_psubsetI,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( A4 != B5 )
=> ( ord_less_set_nat @ A4 @ B5 ) ) ) ).
% psubsetI
thf(fact_175_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_176_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_177_all__not__in__conv,axiom,
! [A4: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A4 ) )
= ( A4 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_178_all__not__in__conv,axiom,
! [A4: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_179_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_180_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_181_Int__iff,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A4 @ B5 ) )
= ( ( member_a @ C @ A4 )
& ( member_a @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_182_Int__iff,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( ( member_nat @ C @ A4 )
& ( member_nat @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_183_IntI,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ A4 )
=> ( ( member_a @ C @ B5 )
=> ( member_a @ C @ ( inf_inf_set_a @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_184_IntI,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ A4 )
=> ( ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_185_atMost__eq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ( set_ord_atMost_nat @ X2 )
= ( set_ord_atMost_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% atMost_eq_iff
thf(fact_186_atLeast__eq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ( set_ord_atLeast_nat @ X2 )
= ( set_ord_atLeast_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% atLeast_eq_iff
thf(fact_187_atMost__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_ord_atMost_a @ K ) )
= ( ord_less_eq_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_188_atMost__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_189_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_190_atLeast__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_ord_atLeast_a @ K ) )
= ( ord_less_eq_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_191_atLeast__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or1731685050470061051et_nat @ K ) )
= ( ord_less_eq_set_nat @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_192_atLeast__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atLeast_nat @ K ) )
= ( ord_less_eq_nat @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_193_disjoint__iff__not__equal,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ! [Y: nat] :
( ( member_nat @ Y @ B5 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_194_not__psubset__empty,axiom,
! [A4: set_nat] :
~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_195_Int__empty__right,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_196_Int__empty__left,axiom,
! [B5: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B5 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_197_disjoint__iff,axiom,
! [A4: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ A4 @ B5 )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A4 )
=> ~ ( member_a @ X @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_198_disjoint__iff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ~ ( member_nat @ X @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_199_ex__in__conv,axiom,
! [A4: set_a] :
( ( ? [X: a] : ( member_a @ X @ A4 ) )
= ( A4 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_200_ex__in__conv,axiom,
! [A4: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A4 ) )
= ( A4 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_201_Int__emptyI,axiom,
! [A4: set_a,B5: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ~ ( member_a @ X3 @ B5 ) )
=> ( ( inf_inf_set_a @ A4 @ B5 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_202_Int__emptyI,axiom,
! [A4: set_nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ~ ( member_nat @ X3 @ B5 ) )
=> ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_203_psubsetD,axiom,
! [A4: set_a,B5: set_a,C: a] :
( ( ord_less_set_a @ A4 @ B5 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_204_psubsetD,axiom,
! [A4: set_nat,B5: set_nat,C: nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_205_equals0I,axiom,
! [A4: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A4 )
=> ( A4 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_206_equals0I,axiom,
! [A4: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A4 )
=> ( A4 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_207_equals0D,axiom,
! [A4: set_a,A: a] :
( ( A4 = bot_bot_set_a )
=> ~ ( member_a @ A @ A4 ) ) ).
% equals0D
thf(fact_208_equals0D,axiom,
! [A4: set_nat,A: nat] :
( ( A4 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_209_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_210_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_211_IntD2,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A4 @ B5 ) )
=> ( member_a @ C @ B5 ) ) ).
% IntD2
thf(fact_212_IntD2,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) )
=> ( member_nat @ C @ B5 ) ) ).
% IntD2
thf(fact_213_IntD1,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A4 @ B5 ) )
=> ( member_a @ C @ A4 ) ) ).
% IntD1
thf(fact_214_IntD1,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) )
=> ( member_nat @ C @ A4 ) ) ).
% IntD1
thf(fact_215_IntE,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A4 @ B5 ) )
=> ~ ( ( member_a @ C @ A4 )
=> ~ ( member_a @ C @ B5 ) ) ) ).
% IntE
thf(fact_216_IntE,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) )
=> ~ ( ( member_nat @ C @ A4 )
=> ~ ( member_nat @ C @ B5 ) ) ) ).
% IntE
thf(fact_217_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_218_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_219_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_220_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_221_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_222_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_223_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_224_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_225_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_226_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_227_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_228_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_229_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_230_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_231_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_232_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_233_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_234_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_235_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_236_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_237_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_238_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_239_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_240_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_241_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_242_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_243_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_244_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_245_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_246_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_247_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_248_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_249_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_250_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_251_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_252_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_253_gt__ex,axiom,
! [X2: nat] :
? [X_12: nat] : ( ord_less_nat @ X2 @ X_12 ) ).
% gt_ex
thf(fact_254_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_255_not__Iic__eq__Ici,axiom,
! [H: nat,L: nat] :
( ( set_ord_atMost_nat @ H )
!= ( set_ord_atLeast_nat @ L ) ) ).
% not_Iic_eq_Ici
thf(fact_256_not__empty__eq__Ici__eq__empty,axiom,
! [L2: nat] :
( bot_bot_set_nat
!= ( set_ord_atLeast_nat @ L2 ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_257_not__empty__eq__Iic__eq__empty,axiom,
! [H: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_258_not__Ici__le__Iic,axiom,
! [L2: nat,H2: nat] :
~ ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ L2 ) @ ( set_ord_atMost_nat @ H2 ) ) ).
% not_Ici_le_Iic
thf(fact_259_order_Opreordering__axioms,axiom,
preordering_a @ ord_less_eq_a @ ord_less_a ).
% order.preordering_axioms
thf(fact_260_order_Opreordering__axioms,axiom,
preordering_nat @ ord_less_eq_nat @ ord_less_nat ).
% order.preordering_axioms
thf(fact_261_order_Opreordering__axioms,axiom,
preordering_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% order.preordering_axioms
thf(fact_262_order_Oordering__axioms,axiom,
ordering_a @ ord_less_eq_a @ ord_less_a ).
% order.ordering_axioms
thf(fact_263_order_Oordering__axioms,axiom,
ordering_nat @ ord_less_eq_nat @ ord_less_nat ).
% order.ordering_axioms
thf(fact_264_order_Oordering__axioms,axiom,
ordering_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% order.ordering_axioms
thf(fact_265_le__sup__lexord,axiom,
! [K: a > a,A4: a,B5: a,Ca: a,C: a,S: a] :
( ( ( ord_less_a @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ B5 ) )
=> ( ( ( ord_less_a @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ord_less_eq_a @ Ca @ A4 ) )
=> ( ( ( ( K @ A4 )
= ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ S ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur6235662355876231836rd_a_a @ A4 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_266_le__sup__lexord,axiom,
! [K: a > nat,A4: a,B5: a,Ca: a,C: a,S: a] :
( ( ( ord_less_nat @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ B5 ) )
=> ( ( ( ord_less_nat @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ord_less_eq_a @ Ca @ A4 ) )
=> ( ( ( ( K @ A4 )
= ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_nat @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ~ ( ord_less_eq_nat @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ S ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur6529588965902446770_a_nat @ A4 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_267_le__sup__lexord,axiom,
! [K: a > set_nat,A4: a,B5: a,Ca: a,C: a,S: a] :
( ( ( ord_less_set_nat @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ B5 ) )
=> ( ( ( ord_less_set_nat @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ord_less_eq_a @ Ca @ A4 ) )
=> ( ( ( ( K @ A4 )
= ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_set_nat @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ~ ( ord_less_eq_set_nat @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ S ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur2251796063117055080et_nat @ A4 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_268_le__sup__lexord,axiom,
! [K: nat > a,A4: nat,B5: nat,Ca: nat,C: nat,S: nat] :
( ( ( ord_less_a @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ B5 ) )
=> ( ( ( ord_less_a @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ord_less_eq_nat @ Ca @ A4 ) )
=> ( ( ( ( K @ A4 )
= ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ S ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur1338148430225957392_nat_a @ A4 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_269_le__sup__lexord,axiom,
! [K: nat > nat,A4: nat,B5: nat,Ca: nat,C: nat,S: nat] :
( ( ( ord_less_nat @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ B5 ) )
=> ( ( ( ord_less_nat @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ord_less_eq_nat @ Ca @ A4 ) )
=> ( ( ( ( K @ A4 )
= ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_nat @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ~ ( ord_less_eq_nat @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ S ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur4601247141005857854at_nat @ A4 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_270_le__sup__lexord,axiom,
! [K: nat > set_nat,A4: nat,B5: nat,Ca: nat,C: nat,S: nat] :
( ( ( ord_less_set_nat @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ B5 ) )
=> ( ( ( ord_less_set_nat @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ord_less_eq_nat @ Ca @ A4 ) )
=> ( ( ( ( K @ A4 )
= ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_set_nat @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ~ ( ord_less_eq_set_nat @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ S ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur6959333727155881972et_nat @ A4 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_271_le__sup__lexord,axiom,
! [K: set_nat > a,A4: set_nat,B5: set_nat,Ca: set_nat,C: set_nat,S: set_nat] :
( ( ( ord_less_a @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_nat @ Ca @ B5 ) )
=> ( ( ( ord_less_a @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ord_less_eq_set_nat @ Ca @ A4 ) )
=> ( ( ( ( K @ A4 )
= ( K @ B5 ) )
=> ( ord_less_eq_set_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_nat @ Ca @ S ) ) )
=> ( ord_less_eq_set_nat @ Ca @ ( measur7612622215237796826_nat_a @ A4 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_272_le__sup__lexord,axiom,
! [K: set_nat > nat,A4: set_nat,B5: set_nat,Ca: set_nat,C: set_nat,S: set_nat] :
( ( ( ord_less_nat @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_nat @ Ca @ B5 ) )
=> ( ( ( ord_less_nat @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ord_less_eq_set_nat @ Ca @ A4 ) )
=> ( ( ( ( K @ A4 )
= ( K @ B5 ) )
=> ( ord_less_eq_set_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_nat @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ~ ( ord_less_eq_nat @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_nat @ Ca @ S ) ) )
=> ( ord_less_eq_set_nat @ Ca @ ( measur3393698822500512756at_nat @ A4 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_273_le__sup__lexord,axiom,
! [K: set_nat > set_nat,A4: set_nat,B5: set_nat,Ca: set_nat,C: set_nat,S: set_nat] :
( ( ( ord_less_set_nat @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_nat @ Ca @ B5 ) )
=> ( ( ( ord_less_set_nat @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ord_less_eq_set_nat @ Ca @ A4 ) )
=> ( ( ( ( K @ A4 )
= ( K @ B5 ) )
=> ( ord_less_eq_set_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_set_nat @ ( K @ B5 ) @ ( K @ A4 ) )
=> ( ~ ( ord_less_eq_set_nat @ ( K @ A4 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_nat @ Ca @ S ) ) )
=> ( ord_less_eq_set_nat @ Ca @ ( measur5257060982548205482et_nat @ A4 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_274_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_275_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_276_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_277_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_278_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_279_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_280_verit__comp__simplify1_I3_J,axiom,
! [B7: a,A6: a] :
( ( ~ ( ord_less_eq_a @ B7 @ A6 ) )
= ( ord_less_a @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_281_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_282_leD,axiom,
! [Y2: a,X2: a] :
( ( ord_less_eq_a @ Y2 @ X2 )
=> ~ ( ord_less_a @ X2 @ Y2 ) ) ).
% leD
thf(fact_283_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_284_leD,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_285_leI,axiom,
! [X2: a,Y2: a] :
( ~ ( ord_less_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% leI
thf(fact_286_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_287_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_288_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_289_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_290_antisym__conv1,axiom,
! [X2: a,Y2: a] :
( ~ ( ord_less_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_291_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_292_antisym__conv1,axiom,
! [X2: set_nat,Y2: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_293_antisym__conv2,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ~ ( ord_less_a @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_294_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_295_antisym__conv2,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_296_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ~ ( ord_less_eq_a @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_297_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_298_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
& ~ ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_299_not__le__imp__less,axiom,
! [Y2: a,X2: a] :
( ~ ( ord_less_eq_a @ Y2 @ X2 )
=> ( ord_less_a @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_300_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_301_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_302_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_303_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_304_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_305_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_306_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_307_order_Ostrict__trans1,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_308_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_309_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_310_order_Ostrict__trans2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_311_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_312_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_313_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ~ ( ord_less_eq_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_314_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_315_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_316_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_317_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_318_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_319_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_320_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_321_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_322_dual__order_Ostrict__trans1,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_323_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_324_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_325_dual__order_Ostrict__trans2,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_326_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_327_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_328_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ~ ( ord_less_eq_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_329_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_330_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ~ ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_331_order_Ostrict__implies__order,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_eq_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_332_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_333_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_334_dual__order_Ostrict__implies__order,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_eq_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_335_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_336_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_337_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_338_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_339_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_340_order__less__le,axiom,
( ord_less_a
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_341_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_342_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_343_linorder__not__le,axiom,
! [X2: a,Y2: a] :
( ( ~ ( ord_less_eq_a @ X2 @ Y2 ) )
= ( ord_less_a @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_344_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_345_linorder__not__less,axiom,
! [X2: a,Y2: a] :
( ( ~ ( ord_less_a @ X2 @ Y2 ) )
= ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_346_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_347_order__less__imp__le,axiom,
! [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_348_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_349_order__less__imp__le,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_350_order__le__neq__trans,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_351_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_352_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_353_order__neq__le__trans,axiom,
! [A: a,B: a] :
( ( A != B )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_354_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_355_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_356_order__le__less__trans,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_a @ Y2 @ Z2 )
=> ( ord_less_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_357_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_358_order__le__less__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_set_nat @ Y2 @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_359_order__less__le__trans,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ( ord_less_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_360_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_361_order__less__le__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_362_order__le__less__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_363_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_364_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_365_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_366_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_367_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_368_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_369_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_370_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_371_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > a,C: a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_372_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_373_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_374_order__less__le__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_375_order__less__le__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_376_order__less__le__subst1,axiom,
! [A: set_nat,F: a > set_nat,B: a,C: a] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_377_order__less__le__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_378_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_379_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_380_order__less__le__subst1,axiom,
! [A: a,F: set_nat > a,B: set_nat,C: set_nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_381_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_382_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_383_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_384_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_385_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_386_linorder__le__less__linear,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
| ( ord_less_a @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_387_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_388_order__le__imp__less__or__eq,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_389_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_390_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_set_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_391_psubsetE,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ~ ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ).
% psubsetE
thf(fact_392_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_393_psubset__imp__subset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_394_psubset__subset__trans,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ord_less_set_nat @ A4 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_395_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B6 )
& ~ ( ord_less_eq_set_nat @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_396_subset__psubset__trans,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_set_nat @ B5 @ C2 )
=> ( ord_less_set_nat @ A4 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_397_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_set_nat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_398_Int__mono,axiom,
! [A4: set_nat,C2: set_nat,B5: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B5 @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_399_Int__lower1,axiom,
! [A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ A4 ) ).
% Int_lower1
thf(fact_400_Int__lower2,axiom,
! [A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_401_Int__absorb1,axiom,
! [B5: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_402_Int__absorb2,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( inf_inf_set_nat @ A4 @ B5 )
= A4 ) ) ).
% Int_absorb2
thf(fact_403_Int__greatest,axiom,
! [C2: set_nat,A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A4 )
=> ( ( ord_less_eq_set_nat @ C2 @ B5 )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_404_Int__Collect__mono,axiom,
! [A4: set_a,B5: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B5 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_405_Int__Collect__mono,axiom,
! [A4: set_nat,B5: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B5 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_406_inf__bot__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_407_inf__bot__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_408_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_409_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_410_le__inf__iff,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_411_le__inf__iff,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X2 @ Y2 )
& ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_412_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_413_inf_Obounded__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A @ B )
& ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_414_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_415_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_416_inf_OcoboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_417_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_418_inf_OcoboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_419_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_420_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_421_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_422_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_423_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_424_inf_Ocobounded2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_425_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_426_inf_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_427_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_428_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( A2
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_429_inf__greatest,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_430_inf__greatest,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_431_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_432_inf_OboundedI,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_433_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_434_inf_OboundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_435_inf__absorb2,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_436_inf__absorb2,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( inf_inf_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_437_inf__absorb1,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_438_inf__absorb1,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( inf_inf_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_439_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_440_inf_Oabsorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_441_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_442_inf_Oabsorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_443_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_444_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_445_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y2: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ X3 @ Z4 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_446_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y2: set_nat] :
( ! [X3: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: set_nat,Y4: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_set_nat @ X3 @ Z4 )
=> ( ord_less_eq_set_nat @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_set_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_447_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_448_inf_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( inf_inf_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_449_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_450_inf_OorderE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( A
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_451_le__infI2,axiom,
! [B: nat,X2: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_452_le__infI2,axiom,
! [B: set_nat,X2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_453_le__infI1,axiom,
! [A: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_454_le__infI1,axiom,
! [A: set_nat,X2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_455_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_456_inf__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_457_le__infI,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A )
=> ( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_458_le__infI,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ B )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_459_le__infE,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A )
=> ~ ( ord_less_eq_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_460_le__infE,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_set_nat @ X2 @ A )
=> ~ ( ord_less_eq_set_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_461_inf__le2,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_462_inf__le2,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_463_inf__le1,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_464_inf__le1,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_465_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_466_inf__sup__ord_I1_J,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_467_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_468_inf__sup__ord_I2_J,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_469_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_470_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_471_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_472_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_473_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_474_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_475_less__infI2,axiom,
! [B: nat,X2: nat,A: nat] :
( ( ord_less_nat @ B @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% less_infI2
thf(fact_476_less__infI1,axiom,
! [A: nat,X2: nat,B: nat] :
( ( ord_less_nat @ A @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% less_infI1
thf(fact_477_subset__emptyI,axiom,
! [A4: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A4 )
=> ( ord_less_eq_set_a @ A4 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_478_subset__emptyI,axiom,
! [A4: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_479_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_480_pinf_I6_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X5: a] :
( ( ord_less_a @ Z4 @ X5 )
=> ~ ( ord_less_eq_a @ X5 @ T2 ) ) ).
% pinf(6)
thf(fact_481_pinf_I6_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T2 ) ) ).
% pinf(6)
thf(fact_482_pinf_I8_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X5: a] :
( ( ord_less_a @ Z4 @ X5 )
=> ( ord_less_eq_a @ T2 @ X5 ) ) ).
% pinf(8)
thf(fact_483_pinf_I8_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_eq_nat @ T2 @ X5 ) ) ).
% pinf(8)
thf(fact_484_minf_I6_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X5: a] :
( ( ord_less_a @ X5 @ Z4 )
=> ( ord_less_eq_a @ X5 @ T2 ) ) ).
% minf(6)
thf(fact_485_minf_I6_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_eq_nat @ X5 @ T2 ) ) ).
% minf(6)
thf(fact_486_minf_I8_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X5: a] :
( ( ord_less_a @ X5 @ Z4 )
=> ~ ( ord_less_eq_a @ T2 @ X5 ) ) ).
% minf(8)
thf(fact_487_minf_I8_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_eq_nat @ T2 @ X5 ) ) ).
% minf(8)
thf(fact_488_minf_I7_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_nat @ T2 @ X5 ) ) ).
% minf(7)
thf(fact_489_minf_I5_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_nat @ X5 @ T2 ) ) ).
% minf(5)
thf(fact_490_minf_I4_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T2 ) ) ).
% minf(4)
thf(fact_491_minf_I3_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T2 ) ) ).
% minf(3)
thf(fact_492_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_493_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_494_pinf_I7_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_nat @ T2 @ X5 ) ) ).
% pinf(7)
thf(fact_495_pinf_I5_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T2 ) ) ).
% pinf(5)
thf(fact_496_pinf_I4_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T2 ) ) ).
% pinf(4)
thf(fact_497_pinf_I3_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T2 ) ) ).
% pinf(3)
thf(fact_498_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_499_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_500_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_501_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X: a] : ( member_a @ X @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_502_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_503_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A5: set_nat] : ( A5 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_504_atMost__Int__atLeast,axiom,
! [N2: nat] :
( ( inf_inf_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( set_ord_atLeast_nat @ N2 ) )
= ( insert_nat @ N2 @ bot_bot_set_nat ) ) ).
% atMost_Int_atLeast
thf(fact_505_Icc__subset__Ici__iff,axiom,
! [L2: a,H: a,L: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ L2 @ H ) @ ( set_ord_atLeast_a @ L ) )
= ( ~ ( ord_less_eq_a @ L2 @ H )
| ( ord_less_eq_a @ L @ L2 ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_506_Icc__subset__Ici__iff,axiom,
! [L2: set_nat,H: set_nat,L: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L2 @ H ) @ ( set_or1731685050470061051et_nat @ L ) )
= ( ~ ( ord_less_eq_set_nat @ L2 @ H )
| ( ord_less_eq_set_nat @ L @ L2 ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_507_Icc__subset__Ici__iff,axiom,
! [L2: nat,H: nat,L: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ H ) @ ( set_ord_atLeast_nat @ L ) )
= ( ~ ( ord_less_eq_nat @ L2 @ H )
| ( ord_less_eq_nat @ L @ L2 ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_508_Icc__subset__Iic__iff,axiom,
! [L2: a,H: a,H2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ L2 @ H ) @ ( set_ord_atMost_a @ H2 ) )
= ( ~ ( ord_less_eq_a @ L2 @ H )
| ( ord_less_eq_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_509_Icc__subset__Iic__iff,axiom,
! [L2: set_nat,H: set_nat,H2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L2 @ H ) @ ( set_or4236626031148496127et_nat @ H2 ) )
= ( ~ ( ord_less_eq_set_nat @ L2 @ H )
| ( ord_less_eq_set_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_510_Icc__subset__Iic__iff,axiom,
! [L2: nat,H: nat,H2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ H ) @ ( set_ord_atMost_nat @ H2 ) )
= ( ~ ( ord_less_eq_nat @ L2 @ H )
| ( ord_less_eq_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_511_inf_Osemilattice__order__axioms,axiom,
semila1248733672344298208er_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).
% inf.semilattice_order_axioms
thf(fact_512_inf_Osemilattice__order__axioms,axiom,
semila2291775939624898198et_nat @ inf_inf_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% inf.semilattice_order_axioms
thf(fact_513_ivl__disj__int__one_I6_J,axiom,
! [L2: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) @ ( set_ord_atLeast_nat @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(6)
thf(fact_514_insert__absorb2,axiom,
! [X2: nat,A4: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A4 ) )
= ( insert_nat @ X2 @ A4 ) ) ).
% insert_absorb2
thf(fact_515_insert__iff,axiom,
! [A: a,B: a,A4: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A4 ) )
= ( ( A = B )
| ( member_a @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_516_insert__iff,axiom,
! [A: nat,B: nat,A4: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
= ( ( A = B )
| ( member_nat @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_517_insertCI,axiom,
! [A: a,B5: set_a,B: a] :
( ( ~ ( member_a @ A @ B5 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B5 ) ) ) ).
% insertCI
thf(fact_518_insertCI,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B5 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_519_Icc__eq__Icc,axiom,
! [L2: a,H: a,L: a,H2: a] :
( ( ( set_or672772299803893939Most_a @ L2 @ H )
= ( set_or672772299803893939Most_a @ L @ H2 ) )
= ( ( ( L2 = L )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_a @ L2 @ H )
& ~ ( ord_less_eq_a @ L @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_520_Icc__eq__Icc,axiom,
! [L2: set_nat,H: set_nat,L: set_nat,H2: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L2 @ H )
= ( set_or4548717258645045905et_nat @ L @ H2 ) )
= ( ( ( L2 = L )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_nat @ L2 @ H )
& ~ ( ord_less_eq_set_nat @ L @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_521_Icc__eq__Icc,axiom,
! [L2: nat,H: nat,L: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L2 @ H )
= ( set_or1269000886237332187st_nat @ L @ H2 ) )
= ( ( ( L2 = L )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L2 @ H )
& ~ ( ord_less_eq_nat @ L @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_522_atLeastAtMost__iff,axiom,
! [I2: a,L2: a,U: a] :
( ( member_a @ I2 @ ( set_or672772299803893939Most_a @ L2 @ U ) )
= ( ( ord_less_eq_a @ L2 @ I2 )
& ( ord_less_eq_a @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_523_atLeastAtMost__iff,axiom,
! [I2: set_nat,L2: set_nat,U: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4548717258645045905et_nat @ L2 @ U ) )
= ( ( ord_less_eq_set_nat @ L2 @ I2 )
& ( ord_less_eq_set_nat @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_524_atLeastAtMost__iff,axiom,
! [I2: nat,L2: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
= ( ( ord_less_eq_nat @ L2 @ I2 )
& ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_525_insert__subset,axiom,
! [X2: a,A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A4 ) @ B5 )
= ( ( member_a @ X2 @ B5 )
& ( ord_less_eq_set_a @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_526_insert__subset,axiom,
! [X2: nat,A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A4 ) @ B5 )
= ( ( member_nat @ X2 @ B5 )
& ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_527_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_528_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_529_Int__insert__right__if1,axiom,
! [A: a,A4: set_a,B5: set_a] :
( ( member_a @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B5 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A4 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_530_Int__insert__right__if1,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_531_Int__insert__right__if0,axiom,
! [A: a,A4: set_a,B5: set_a] :
( ~ ( member_a @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B5 ) )
= ( inf_inf_set_a @ A4 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_532_Int__insert__right__if0,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ~ ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_533_insert__inter__insert,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ).
% insert_inter_insert
thf(fact_534_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B5: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B5 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_535_Int__insert__left__if1,axiom,
! [A: nat,C2: set_nat,B5: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_536_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B5: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B5 ) @ C2 )
= ( inf_inf_set_a @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_537_Int__insert__left__if0,axiom,
! [A: nat,C2: set_nat,B5: set_nat] :
( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( inf_inf_set_nat @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_538_greaterThanLessThan__iff,axiom,
! [I2: a,L2: a,U: a] :
( ( member_a @ I2 @ ( set_or5939364468397584554Than_a @ L2 @ U ) )
= ( ( ord_less_a @ L2 @ I2 )
& ( ord_less_a @ I2 @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_539_greaterThanLessThan__iff,axiom,
! [I2: nat,L2: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
= ( ( ord_less_nat @ L2 @ I2 )
& ( ord_less_nat @ I2 @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_540_atLeastatMost__subset__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A @ B ) @ ( set_or672772299803893939Most_a @ C @ D2 ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_541_atLeastatMost__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D2 ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_542_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_543_atLeastatMost__empty_H,axiom,
! [A: a,B: a] :
( ~ ( ord_less_eq_a @ A @ B )
=> ( ( set_or672772299803893939Most_a @ A @ B )
= bot_bot_set_a ) ) ).
% atLeastatMost_empty'
thf(fact_544_atLeastatMost__empty_H,axiom,
! [A: set_nat,B: set_nat] :
( ~ ( ord_less_eq_set_nat @ A @ B )
=> ( ( set_or4548717258645045905et_nat @ A @ B )
= bot_bot_set_set_nat ) ) ).
% atLeastatMost_empty'
thf(fact_545_atLeastatMost__empty_H,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty'
thf(fact_546_atLeastatMost__empty__iff2,axiom,
! [A: a,B: a] :
( ( bot_bot_set_a
= ( set_or672772299803893939Most_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_547_atLeastatMost__empty__iff2,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat
= ( set_or4548717258645045905et_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_548_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_549_atLeastatMost__empty__iff,axiom,
! [A: a,B: a] :
( ( ( set_or672772299803893939Most_a @ A @ B )
= bot_bot_set_a )
= ( ~ ( ord_less_eq_a @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_550_atLeastatMost__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( set_or4548717258645045905et_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_551_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_552_singleton__insert__inj__eq_H,axiom,
! [A: nat,A4: set_nat,B: nat] :
( ( ( insert_nat @ A @ A4 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_553_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A4: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_554_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_555_insert__disjoint_I1_J,axiom,
! [A: a,A4: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A4 ) @ B5 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B5 )
& ( ( inf_inf_set_a @ A4 @ B5 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_556_insert__disjoint_I1_J,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ B5 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B5 )
& ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_557_insert__disjoint_I2_J,axiom,
! [A: a,A4: set_a,B5: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A4 ) @ B5 ) )
= ( ~ ( member_a @ A @ B5 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A4 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_558_insert__disjoint_I2_J,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ B5 ) )
= ( ~ ( member_nat @ A @ B5 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_559_disjoint__insert_I1_J,axiom,
! [B5: set_a,A: a,A4: set_a] :
( ( ( inf_inf_set_a @ B5 @ ( insert_a @ A @ A4 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B5 )
& ( ( inf_inf_set_a @ B5 @ A4 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_560_disjoint__insert_I1_J,axiom,
! [B5: set_nat,A: nat,A4: set_nat] :
( ( ( inf_inf_set_nat @ B5 @ ( insert_nat @ A @ A4 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B5 )
& ( ( inf_inf_set_nat @ B5 @ A4 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_561_disjoint__insert_I2_J,axiom,
! [A4: set_a,B: a,B5: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A4 @ ( insert_a @ B @ B5 ) ) )
= ( ~ ( member_a @ B @ A4 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A4 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_562_disjoint__insert_I2_J,axiom,
! [A4: set_nat,B: nat,B5: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A4 @ ( insert_nat @ B @ B5 ) ) )
= ( ~ ( member_nat @ B @ A4 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_563_atLeastAtMost__singleton,axiom,
! [A: nat] :
( ( set_or1269000886237332187st_nat @ A @ A )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% atLeastAtMost_singleton
thf(fact_564_atLeastAtMost__singleton__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ C @ bot_bot_set_nat ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_565_greaterThanLessThan__empty,axiom,
! [L2: a,K: a] :
( ( ord_less_eq_a @ L2 @ K )
=> ( ( set_or5939364468397584554Than_a @ K @ L2 )
= bot_bot_set_a ) ) ).
% greaterThanLessThan_empty
thf(fact_566_greaterThanLessThan__empty,axiom,
! [L2: set_nat,K: set_nat] :
( ( ord_less_eq_set_nat @ L2 @ K )
=> ( ( set_or8625682525731655386et_nat @ K @ L2 )
= bot_bot_set_set_nat ) ) ).
% greaterThanLessThan_empty
thf(fact_567_greaterThanLessThan__empty,axiom,
! [L2: nat,K: nat] :
( ( ord_less_eq_nat @ L2 @ K )
=> ( ( set_or5834768355832116004an_nat @ K @ L2 )
= bot_bot_set_nat ) ) ).
% greaterThanLessThan_empty
thf(fact_568_mk__disjoint__insert,axiom,
! [A: a,A4: set_a] :
( ( member_a @ A @ A4 )
=> ? [B8: set_a] :
( ( A4
= ( insert_a @ A @ B8 ) )
& ~ ( member_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_569_mk__disjoint__insert,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ? [B8: set_nat] :
( ( A4
= ( insert_nat @ A @ B8 ) )
& ~ ( member_nat @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_570_insert__commute,axiom,
! [X2: nat,Y2: nat,A4: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ Y2 @ A4 ) )
= ( insert_nat @ Y2 @ ( insert_nat @ X2 @ A4 ) ) ) ).
% insert_commute
thf(fact_571_insert__eq__iff,axiom,
! [A: a,A4: set_a,B: a,B5: set_a] :
( ~ ( member_a @ A @ A4 )
=> ( ~ ( member_a @ B @ B5 )
=> ( ( ( insert_a @ A @ A4 )
= ( insert_a @ B @ B5 ) )
= ( ( ( A = B )
=> ( A4 = B5 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A4
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B5
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_572_insert__eq__iff,axiom,
! [A: nat,A4: set_nat,B: nat,B5: set_nat] :
( ~ ( member_nat @ A @ A4 )
=> ( ~ ( member_nat @ B @ B5 )
=> ( ( ( insert_nat @ A @ A4 )
= ( insert_nat @ B @ B5 ) )
= ( ( ( A = B )
=> ( A4 = B5 ) )
& ( ( A != B )
=> ? [C4: set_nat] :
( ( A4
= ( insert_nat @ B @ C4 ) )
& ~ ( member_nat @ B @ C4 )
& ( B5
= ( insert_nat @ A @ C4 ) )
& ~ ( member_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_573_insert__absorb,axiom,
! [A: a,A4: set_a] :
( ( member_a @ A @ A4 )
=> ( ( insert_a @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_574_insert__absorb,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( insert_nat @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_575_insert__ident,axiom,
! [X2: a,A4: set_a,B5: set_a] :
( ~ ( member_a @ X2 @ A4 )
=> ( ~ ( member_a @ X2 @ B5 )
=> ( ( ( insert_a @ X2 @ A4 )
= ( insert_a @ X2 @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_576_insert__ident,axiom,
! [X2: nat,A4: set_nat,B5: set_nat] :
( ~ ( member_nat @ X2 @ A4 )
=> ( ~ ( member_nat @ X2 @ B5 )
=> ( ( ( insert_nat @ X2 @ A4 )
= ( insert_nat @ X2 @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_577_Set_Oset__insert,axiom,
! [X2: a,A4: set_a] :
( ( member_a @ X2 @ A4 )
=> ~ ! [B8: set_a] :
( ( A4
= ( insert_a @ X2 @ B8 ) )
=> ( member_a @ X2 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_578_Set_Oset__insert,axiom,
! [X2: nat,A4: set_nat] :
( ( member_nat @ X2 @ A4 )
=> ~ ! [B8: set_nat] :
( ( A4
= ( insert_nat @ X2 @ B8 ) )
=> ( member_nat @ X2 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_579_insertI2,axiom,
! [A: a,B5: set_a,B: a] :
( ( member_a @ A @ B5 )
=> ( member_a @ A @ ( insert_a @ B @ B5 ) ) ) ).
% insertI2
thf(fact_580_insertI2,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( member_nat @ A @ B5 )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertI2
thf(fact_581_insertI1,axiom,
! [A: a,B5: set_a] : ( member_a @ A @ ( insert_a @ A @ B5 ) ) ).
% insertI1
thf(fact_582_insertI1,axiom,
! [A: nat,B5: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_583_insertE,axiom,
! [A: a,B: a,A4: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A4 ) )
=> ( ( A != B )
=> ( member_a @ A @ A4 ) ) ) ).
% insertE
thf(fact_584_insertE,axiom,
! [A: nat,B: nat,A4: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A4 ) ) ) ).
% insertE
thf(fact_585_ivl__disj__int__two_I4_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ M4 ) @ ( set_or5834768355832116004an_nat @ M4 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(4)
thf(fact_586_ivl__disj__int__two_I5_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or5834768355832116004an_nat @ L2 @ M4 ) @ ( set_or1269000886237332187st_nat @ M4 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(5)
thf(fact_587_atLeastAtMost__singleton_H,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_588_insert__subsetI,axiom,
! [X2: a,A4: set_a,X7: set_a] :
( ( member_a @ X2 @ A4 )
=> ( ( ord_less_eq_set_a @ X7 @ A4 )
=> ( ord_less_eq_set_a @ ( insert_a @ X2 @ X7 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_589_insert__subsetI,axiom,
! [X2: nat,A4: set_nat,X7: set_nat] :
( ( member_nat @ X2 @ A4 )
=> ( ( ord_less_eq_set_nat @ X7 @ A4 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X7 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_590_insert__mono,axiom,
! [C2: set_nat,D: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_591_subset__insert,axiom,
! [X2: a,A4: set_a,B5: set_a] :
( ~ ( member_a @ X2 @ A4 )
=> ( ( ord_less_eq_set_a @ A4 @ ( insert_a @ X2 @ B5 ) )
= ( ord_less_eq_set_a @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_592_subset__insert,axiom,
! [X2: nat,A4: set_nat,B5: set_nat] :
( ~ ( member_nat @ X2 @ A4 )
=> ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X2 @ B5 ) )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_593_subset__insertI,axiom,
! [B5: set_nat,A: nat] : ( ord_less_eq_set_nat @ B5 @ ( insert_nat @ A @ B5 ) ) ).
% subset_insertI
thf(fact_594_subset__insertI2,axiom,
! [A4: set_nat,B5: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ B5 ) ) ) ).
% subset_insertI2
thf(fact_595_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_596_insert__not__empty,axiom,
! [A: nat,A4: set_nat] :
( ( insert_nat @ A @ A4 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_597_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_598_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_599_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_600_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_601_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_602_not__Ici__eq__Icc,axiom,
! [L: nat,L2: nat,H: nat] :
( ( set_ord_atLeast_nat @ L )
!= ( set_or1269000886237332187st_nat @ L2 @ H ) ) ).
% not_Ici_eq_Icc
thf(fact_603_Int__insert__right,axiom,
! [A: a,A4: set_a,B5: set_a] :
( ( ( member_a @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B5 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A4 @ B5 ) ) ) )
& ( ~ ( member_a @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B5 ) )
= ( inf_inf_set_a @ A4 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_604_Int__insert__right,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) )
& ( ~ ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_605_Int__insert__left,axiom,
! [A: a,C2: set_a,B5: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B5 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B5 @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B5 ) @ C2 )
= ( inf_inf_set_a @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_606_Int__insert__left,axiom,
! [A: nat,C2: set_nat,B5: set_nat] :
( ( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B5 @ C2 ) ) ) )
& ( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( inf_inf_set_nat @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_607_not__Ici__le__Icc,axiom,
! [L2: nat,L: nat,H2: nat] :
~ ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ L2 ) @ ( set_or1269000886237332187st_nat @ L @ H2 ) ) ).
% not_Ici_le_Icc
thf(fact_608_subset__singleton__iff,axiom,
! [X7: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X7 = bot_bot_set_nat )
| ( X7
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_609_subset__singletonD,axiom,
! [A4: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ( A4 = bot_bot_set_nat )
| ( A4
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_610_atLeastatMost__psubset__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_set_a @ ( set_or672772299803893939Most_a @ A @ B ) @ ( set_or672772299803893939Most_a @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_a @ A @ B )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D2 )
& ( ( ord_less_a @ C @ A )
| ( ord_less_a @ B @ D2 ) ) ) )
& ( ord_less_eq_a @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_611_atLeastatMost__psubset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D2 )
& ( ( ord_less_set_nat @ C @ A )
| ( ord_less_set_nat @ B @ D2 ) ) ) )
& ( ord_less_eq_set_nat @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_612_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D2 )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D2 ) ) ) )
& ( ord_less_eq_nat @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_613_atLeastAtMost__def,axiom,
( set_or1269000886237332187st_nat
= ( ^ [L3: nat,U2: nat] : ( inf_inf_set_nat @ ( set_ord_atLeast_nat @ L3 ) @ ( set_ord_atMost_nat @ U2 ) ) ) ) ).
% atLeastAtMost_def
thf(fact_614_ivl__disj__int__one_I1_J,axiom,
! [L2: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_ord_atMost_nat @ L2 ) @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(1)
thf(fact_615_the__elem__eq,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_616_is__singletonI,axiom,
! [X2: nat] : ( is_singleton_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_617_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A5: set_nat] :
? [X: nat] :
( A5
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_618_is__singletonE,axiom,
! [A4: set_nat] :
( ( is_singleton_nat @ A4 )
=> ~ ! [X3: nat] :
( A4
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_619_atLeastAtMost__diff__ends,axiom,
! [A: nat,B: nat] :
( ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( set_or5834768355832116004an_nat @ A @ B ) ) ).
% atLeastAtMost_diff_ends
thf(fact_620_psubset__insert__iff,axiom,
! [A4: set_a,X2: a,B5: set_a] :
( ( ord_less_set_a @ A4 @ ( insert_a @ X2 @ B5 ) )
= ( ( ( member_a @ X2 @ B5 )
=> ( ord_less_set_a @ A4 @ B5 ) )
& ( ~ ( member_a @ X2 @ B5 )
=> ( ( ( member_a @ X2 @ A4 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B5 ) )
& ( ~ ( member_a @ X2 @ A4 )
=> ( ord_less_eq_set_a @ A4 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_621_psubset__insert__iff,axiom,
! [A4: set_nat,X2: nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ ( insert_nat @ X2 @ B5 ) )
= ( ( ( member_nat @ X2 @ B5 )
=> ( ord_less_set_nat @ A4 @ B5 ) )
& ( ~ ( member_nat @ X2 @ B5 )
=> ( ( ( member_nat @ X2 @ A4 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B5 ) )
& ( ~ ( member_nat @ X2 @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_622_DiffI,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ A4 )
=> ( ~ ( member_a @ C @ B5 )
=> ( member_a @ C @ ( minus_minus_set_a @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_623_DiffI,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ A4 )
=> ( ~ ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_624_Diff__iff,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A4 @ B5 ) )
= ( ( member_a @ C @ A4 )
& ~ ( member_a @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_625_Diff__iff,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( ( member_nat @ C @ A4 )
& ~ ( member_nat @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_626_Diff__empty,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ bot_bot_set_nat )
= A4 ) ).
% Diff_empty
thf(fact_627_empty__Diff,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A4 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_628_Diff__cancel,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ A4 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_629_Diff__insert0,axiom,
! [X2: a,A4: set_a,B5: set_a] :
( ~ ( member_a @ X2 @ A4 )
=> ( ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ B5 ) )
= ( minus_minus_set_a @ A4 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_630_Diff__insert0,axiom,
! [X2: nat,A4: set_nat,B5: set_nat] :
( ~ ( member_nat @ X2 @ A4 )
=> ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ B5 ) )
= ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_631_insert__Diff1,axiom,
! [X2: a,B5: set_a,A4: set_a] :
( ( member_a @ X2 @ B5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A4 ) @ B5 )
= ( minus_minus_set_a @ A4 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_632_insert__Diff1,axiom,
! [X2: nat,B5: set_nat,A4: set_nat] :
( ( member_nat @ X2 @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A4 ) @ B5 )
= ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_633_Diff__eq__empty__iff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( minus_minus_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_634_insert__Diff__single,axiom,
! [A: nat,A4: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_635_Diff__disjoint,axiom,
! [A4: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ A4 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_636_DiffE,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A4 @ B5 ) )
=> ~ ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B5 ) ) ) ).
% DiffE
thf(fact_637_DiffE,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) )
=> ~ ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% DiffE
thf(fact_638_DiffD1,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A4 @ B5 ) )
=> ( member_a @ C @ A4 ) ) ).
% DiffD1
thf(fact_639_DiffD1,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) )
=> ( member_nat @ C @ A4 ) ) ).
% DiffD1
thf(fact_640_DiffD2,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A4 @ B5 ) )
=> ~ ( member_a @ C @ B5 ) ) ).
% DiffD2
thf(fact_641_DiffD2,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) )
=> ~ ( member_nat @ C @ B5 ) ) ).
% DiffD2
thf(fact_642_double__diff,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ( minus_minus_set_nat @ B5 @ ( minus_minus_set_nat @ C2 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_643_Diff__subset,axiom,
! [A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ A4 ) ).
% Diff_subset
thf(fact_644_Diff__mono,axiom,
! [A4: set_nat,C2: set_nat,D: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ D @ B5 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ ( minus_minus_set_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_645_insert__Diff__if,axiom,
! [X2: a,B5: set_a,A4: set_a] :
( ( ( member_a @ X2 @ B5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A4 ) @ B5 )
= ( minus_minus_set_a @ A4 @ B5 ) ) )
& ( ~ ( member_a @ X2 @ B5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A4 ) @ B5 )
= ( insert_a @ X2 @ ( minus_minus_set_a @ A4 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_646_insert__Diff__if,axiom,
! [X2: nat,B5: set_nat,A4: set_nat] :
( ( ( member_nat @ X2 @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A4 ) @ B5 )
= ( minus_minus_set_nat @ A4 @ B5 ) ) )
& ( ~ ( member_nat @ X2 @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A4 ) @ B5 )
= ( insert_nat @ X2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_647_psubset__imp__ex__mem,axiom,
! [A4: set_a,B5: set_a] :
( ( ord_less_set_a @ A4 @ B5 )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_648_psubset__imp__ex__mem,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_649_diff__shunt__var,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_650_subset__Diff__insert,axiom,
! [A4: set_a,B5: set_a,X2: a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( minus_minus_set_a @ B5 @ ( insert_a @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A4 @ ( minus_minus_set_a @ B5 @ C2 ) )
& ~ ( member_a @ X2 @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_651_subset__Diff__insert,axiom,
! [A4: set_nat,B5: set_nat,X2: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ ( insert_nat @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ C2 ) )
& ~ ( member_nat @ X2 @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_652_Diff__insert__absorb,axiom,
! [X2: a,A4: set_a] :
( ~ ( member_a @ X2 @ A4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A4 ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_653_Diff__insert__absorb,axiom,
! [X2: nat,A4: set_nat] :
( ~ ( member_nat @ X2 @ A4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A4 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_654_Diff__insert2,axiom,
! [A4: set_nat,A: nat,B5: set_nat] :
( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_655_insert__Diff,axiom,
! [A: a,A4: set_a] :
( ( member_a @ A @ A4 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A4 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_656_insert__Diff,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_657_Diff__insert,axiom,
! [A4: set_nat,A: nat,B5: set_nat] :
( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_658_Diff__triv,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A4 @ B5 )
= A4 ) ) ).
% Diff_triv
thf(fact_659_Int__Diff__disjoint,axiom,
! [A4: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ ( minus_minus_set_nat @ A4 @ B5 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_660_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A5: set_nat] :
( A5
= ( insert_nat @ ( the_elem_nat @ A5 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_661_is__singletonI_H,axiom,
! [A4: set_a] :
( ( A4 != bot_bot_set_a )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ A4 )
=> ( ( member_a @ Y4 @ A4 )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_a @ A4 ) ) ) ).
% is_singletonI'
thf(fact_662_is__singletonI_H,axiom,
! [A4: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat @ Y4 @ A4 )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_nat @ A4 ) ) ) ).
% is_singletonI'
thf(fact_663_subset__insert__iff,axiom,
! [A4: set_a,X2: a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( insert_a @ X2 @ B5 ) )
= ( ( ( member_a @ X2 @ A4 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B5 ) )
& ( ~ ( member_a @ X2 @ A4 )
=> ( ord_less_eq_set_a @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_664_subset__insert__iff,axiom,
! [A4: set_nat,X2: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X2 @ B5 ) )
= ( ( ( member_nat @ X2 @ A4 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B5 ) )
& ( ~ ( member_nat @ X2 @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_665_Diff__single__insert,axiom,
! [A4: set_nat,X2: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B5 )
=> ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X2 @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_666_remove__def,axiom,
( remove_nat
= ( ^ [X: nat,A5: set_nat] : ( minus_minus_set_nat @ A5 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_667_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
( set_or6659071591806873216st_nat
= ( ^ [A2: nat,B2: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_668_image__eqI,axiom,
! [B: a,F: a > a,X2: a,A4: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A4 )
=> ( member_a @ B @ ( image_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_669_image__eqI,axiom,
! [B: nat,F: a > nat,X2: a,A4: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A4 )
=> ( member_nat @ B @ ( image_a_nat @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_670_image__eqI,axiom,
! [B: a,F: nat > a,X2: nat,A4: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( member_a @ B @ ( image_nat_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_671_image__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,A4: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_672_member__remove,axiom,
! [X2: a,Y2: a,A4: set_a] :
( ( member_a @ X2 @ ( remove_a @ Y2 @ A4 ) )
= ( ( member_a @ X2 @ A4 )
& ( X2 != Y2 ) ) ) ).
% member_remove
thf(fact_673_member__remove,axiom,
! [X2: nat,Y2: nat,A4: set_nat] :
( ( member_nat @ X2 @ ( remove_nat @ Y2 @ A4 ) )
= ( ( member_nat @ X2 @ A4 )
& ( X2 != Y2 ) ) ) ).
% member_remove
thf(fact_674_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_675_empty__is__image,axiom,
! [F: nat > nat,A4: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_676_image__is__empty,axiom,
! [F: nat > nat,A4: set_nat] :
( ( ( image_nat_nat @ F @ A4 )
= bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_677_image__insert,axiom,
! [F: nat > nat,A: nat,B5: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B5 ) ) ) ).
% image_insert
thf(fact_678_insert__image,axiom,
! [X2: a,A4: set_a,F: a > nat] :
( ( member_a @ X2 @ A4 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_a_nat @ F @ A4 ) )
= ( image_a_nat @ F @ A4 ) ) ) ).
% insert_image
thf(fact_679_insert__image,axiom,
! [X2: nat,A4: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A4 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A4 ) )
= ( image_nat_nat @ F @ A4 ) ) ) ).
% insert_image
thf(fact_680_greaterThanAtMost__iff,axiom,
! [I2: a,L2: a,U: a] :
( ( member_a @ I2 @ ( set_or4472690218693186638Most_a @ L2 @ U ) )
= ( ( ord_less_a @ L2 @ I2 )
& ( ord_less_eq_a @ I2 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_681_greaterThanAtMost__iff,axiom,
! [I2: set_nat,L2: set_nat,U: set_nat] :
( ( member_set_nat @ I2 @ ( set_or7074010630789208630et_nat @ L2 @ U ) )
= ( ( ord_less_set_nat @ L2 @ I2 )
& ( ord_less_eq_set_nat @ I2 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_682_greaterThanAtMost__iff,axiom,
! [I2: nat,L2: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
= ( ( ord_less_nat @ L2 @ I2 )
& ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_683_greaterThanAtMost__empty,axiom,
! [L2: a,K: a] :
( ( ord_less_eq_a @ L2 @ K )
=> ( ( set_or4472690218693186638Most_a @ K @ L2 )
= bot_bot_set_a ) ) ).
% greaterThanAtMost_empty
thf(fact_684_greaterThanAtMost__empty,axiom,
! [L2: set_nat,K: set_nat] :
( ( ord_less_eq_set_nat @ L2 @ K )
=> ( ( set_or7074010630789208630et_nat @ K @ L2 )
= bot_bot_set_set_nat ) ) ).
% greaterThanAtMost_empty
thf(fact_685_greaterThanAtMost__empty,axiom,
! [L2: nat,K: nat] :
( ( ord_less_eq_nat @ L2 @ K )
=> ( ( set_or6659071591806873216st_nat @ K @ L2 )
= bot_bot_set_nat ) ) ).
% greaterThanAtMost_empty
thf(fact_686_greaterThanAtMost__empty__iff2,axiom,
! [K: nat,L2: nat] :
( ( bot_bot_set_nat
= ( set_or6659071591806873216st_nat @ K @ L2 ) )
= ( ~ ( ord_less_nat @ K @ L2 ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_687_greaterThanAtMost__empty__iff,axiom,
! [K: nat,L2: nat] :
( ( ( set_or6659071591806873216st_nat @ K @ L2 )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ K @ L2 ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_688_imageI,axiom,
! [X2: a,A4: set_a,F: a > a] :
( ( member_a @ X2 @ A4 )
=> ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_689_imageI,axiom,
! [X2: a,A4: set_a,F: a > nat] :
( ( member_a @ X2 @ A4 )
=> ( member_nat @ ( F @ X2 ) @ ( image_a_nat @ F @ A4 ) ) ) ).
% imageI
thf(fact_690_imageI,axiom,
! [X2: nat,A4: set_nat,F: nat > a] :
( ( member_nat @ X2 @ A4 )
=> ( member_a @ ( F @ X2 ) @ ( image_nat_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_691_imageI,axiom,
! [X2: nat,A4: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A4 )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A4 ) ) ) ).
% imageI
thf(fact_692_rev__image__eqI,axiom,
! [X2: a,A4: set_a,B: a,F: a > a] :
( ( member_a @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_693_rev__image__eqI,axiom,
! [X2: a,A4: set_a,B: nat,F: a > nat] :
( ( member_a @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_a_nat @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_694_rev__image__eqI,axiom,
! [X2: nat,A4: set_nat,B: a,F: nat > a] :
( ( member_nat @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_nat_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_695_rev__image__eqI,axiom,
! [X2: nat,A4: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_696_Ioc__inj,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( set_or4472690218693186638Most_a @ A @ B )
= ( set_or4472690218693186638Most_a @ C @ D2 ) )
= ( ( ( ord_less_eq_a @ B @ A )
& ( ord_less_eq_a @ D2 @ C ) )
| ( ( A = C )
& ( B = D2 ) ) ) ) ).
% Ioc_inj
thf(fact_697_Ioc__inj,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( set_or6659071591806873216st_nat @ A @ B )
= ( set_or6659071591806873216st_nat @ C @ D2 ) )
= ( ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ D2 @ C ) )
| ( ( A = C )
& ( B = D2 ) ) ) ) ).
% Ioc_inj
thf(fact_698_image__mono,axiom,
! [A4: set_nat,B5: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B5 ) ) ) ).
% image_mono
thf(fact_699_image__subsetI,axiom,
! [A4: set_a,F: a > a,B5: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ ( F @ X3 ) @ B5 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_700_image__subsetI,axiom,
! [A4: set_nat,F: nat > a,B5: set_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_a @ ( F @ X3 ) @ B5 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_701_image__subsetI,axiom,
! [A4: set_a,F: a > nat,B5: set_nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_nat @ ( F @ X3 ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_702_image__subsetI,axiom,
! [A4: set_nat,F: nat > nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ ( F @ X3 ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_703_subset__imageE,axiom,
! [B5: set_nat,F: nat > nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A4 )
=> ( B5
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_704_subset__image__iff,axiom,
! [B5: set_nat,F: nat > nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A4 )
& ( B5
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_705_Ioc__subset__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ A @ B ) @ ( set_or4472690218693186638Most_a @ C @ D2 ) )
= ( ( ord_less_eq_a @ B @ A )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D2 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_706_Ioc__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ A @ B ) @ ( set_or6659071591806873216st_nat @ C @ D2 ) )
= ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_707_ivl__disj__int__two_I6_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ L2 @ M4 ) @ ( set_or6659071591806873216st_nat @ M4 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(6)
thf(fact_708_in__image__insert__iff,axiom,
! [B5: set_set_a,X2: a,A4: set_a] :
( ! [C5: set_a] :
( ( member_set_a @ C5 @ B5 )
=> ~ ( member_a @ X2 @ C5 ) )
=> ( ( member_set_a @ A4 @ ( image_set_a_set_a @ ( insert_a @ X2 ) @ B5 ) )
= ( ( member_a @ X2 @ A4 )
& ( member_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_709_in__image__insert__iff,axiom,
! [B5: set_set_nat,X2: nat,A4: set_nat] :
( ! [C5: set_nat] :
( ( member_set_nat @ C5 @ B5 )
=> ~ ( member_nat @ X2 @ C5 ) )
=> ( ( member_set_nat @ A4 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X2 ) @ B5 ) )
= ( ( member_nat @ X2 @ A4 )
& ( member_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_710_Ioc__disjoint,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ A @ B ) @ ( set_or4472690218693186638Most_a @ C @ D2 ) )
= bot_bot_set_a )
= ( ( ord_less_eq_a @ B @ A )
| ( ord_less_eq_a @ D2 @ C )
| ( ord_less_eq_a @ B @ C )
| ( ord_less_eq_a @ D2 @ A ) ) ) ).
% Ioc_disjoint
thf(fact_711_Ioc__disjoint,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ A @ B ) @ ( set_or6659071591806873216st_nat @ C @ D2 ) )
= bot_bot_set_nat )
= ( ( ord_less_eq_nat @ B @ A )
| ( ord_less_eq_nat @ D2 @ C )
| ( ord_less_eq_nat @ B @ C )
| ( ord_less_eq_nat @ D2 @ A ) ) ) ).
% Ioc_disjoint
thf(fact_712_ivl__disj__int__two_I8_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ M4 ) @ ( set_or6659071591806873216st_nat @ M4 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(8)
thf(fact_713_ivl__disj__int__one_I3_J,axiom,
! [L2: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_ord_atMost_nat @ L2 ) @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(3)
thf(fact_714_ivl__disj__int__two_I2_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ L2 @ M4 ) @ ( set_or5834768355832116004an_nat @ M4 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(2)
thf(fact_715_all__subset__image,axiom,
! [F: nat > nat,A4: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A4 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_716_ivl__disj__un__singleton_I4_J,axiom,
! [L2: nat,U: nat] :
( ( ord_less_nat @ L2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
= ( set_or6659071591806873216st_nat @ L2 @ U ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_717_ivl__disj__un__two_I5_J,axiom,
! [L2: a,M4: a,U: a] :
( ( ord_less_a @ L2 @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L2 @ M4 ) @ ( set_or672772299803893939Most_a @ M4 @ U ) )
= ( set_or4472690218693186638Most_a @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_718_ivl__disj__un__two_I5_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( ord_less_nat @ L2 @ M4 )
=> ( ( ord_less_eq_nat @ M4 @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L2 @ M4 ) @ ( set_or1269000886237332187st_nat @ M4 @ U ) )
= ( set_or6659071591806873216st_nat @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_719_ivl__disj__un__singleton_I5_J,axiom,
! [L2: a,U: a] :
( ( ord_less_eq_a @ L2 @ U )
=> ( ( sup_sup_set_a @ ( insert_a @ L2 @ bot_bot_set_a ) @ ( set_or4472690218693186638Most_a @ L2 @ U ) )
= ( set_or672772299803893939Most_a @ L2 @ U ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_720_ivl__disj__un__singleton_I5_J,axiom,
! [L2: nat,U: nat] :
( ( ord_less_eq_nat @ L2 @ U )
=> ( ( sup_sup_set_nat @ ( insert_nat @ L2 @ bot_bot_set_nat ) @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
= ( set_or1269000886237332187st_nat @ L2 @ U ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_721_Un__iff,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A4 @ B5 ) )
= ( ( member_a @ C @ A4 )
| ( member_a @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_722_Un__iff,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( ( member_nat @ C @ A4 )
| ( member_nat @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_723_UnCI,axiom,
! [C: a,B5: set_a,A4: set_a] :
( ( ~ ( member_a @ C @ B5 )
=> ( member_a @ C @ A4 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_724_UnCI,axiom,
! [C: nat,B5: set_nat,A4: set_nat] :
( ( ~ ( member_nat @ C @ B5 )
=> ( member_nat @ C @ A4 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_725_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_726_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_727_le__sup__iff,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_nat @ X2 @ Z2 )
& ( ord_less_eq_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_728_le__sup__iff,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X2 @ Z2 )
& ( ord_less_eq_set_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_729_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_730_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_731_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_732_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_733_sup__eq__bot__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y2 )
= bot_bot_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_734_bot__eq__sup__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_735_sup__bot__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_736_sup__bot__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_737_Un__subset__iff,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B5 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A4 @ C2 )
& ( ord_less_eq_set_nat @ B5 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_738_Un__empty,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( sup_sup_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ( A4 = bot_bot_set_nat )
& ( B5 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_739_Un__insert__right,axiom,
! [A4: set_nat,A: nat,B5: set_nat] :
( ( sup_sup_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).
% Un_insert_right
thf(fact_740_Un__insert__left,axiom,
! [A: nat,B5: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B5 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_741_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_742_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_743_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_744_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_745_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_746_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_747_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_748_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_749_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_750_sup_Ocobounded2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_751_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_752_sup_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_753_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_754_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( A2
= ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_755_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_756_sup_OboundedI,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_757_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_758_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_759_sup__absorb2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_760_sup__absorb2,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( sup_sup_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_761_sup__absorb1,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_762_sup__absorb1,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( sup_sup_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_763_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_764_sup_Oabsorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_765_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_766_sup_Oabsorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_767_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y2: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_nat @ Z4 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_768_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y2: set_nat] :
( ! [X3: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_nat,Y4: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z4 @ X3 )
=> ( ord_less_eq_set_nat @ ( F @ Y4 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_set_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_769_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_770_sup_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_771_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_772_sup_OorderE,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_773_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_774_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_775_sup__least,axiom,
! [Y2: nat,X2: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_776_sup__least,axiom,
! [Y2: set_nat,X2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y2 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_777_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_778_sup__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_779_sup_Omono,axiom,
! [C: nat,A: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_780_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_781_le__supI2,axiom,
! [X2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_782_le__supI2,axiom,
! [X2: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ B )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_783_le__supI1,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_784_le__supI1,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_785_sup__ge2,axiom,
! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_786_sup__ge2,axiom,
! [Y2: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_787_sup__ge1,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_788_sup__ge1,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_789_le__supI,axiom,
! [A: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X2 )
=> ( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_790_le__supI,axiom,
! [A: set_nat,X2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X2 )
=> ( ( ord_less_eq_set_nat @ B @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_791_le__supE,axiom,
! [A: nat,B: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A @ X2 )
=> ~ ( ord_less_eq_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_792_le__supE,axiom,
! [A: set_nat,B: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ X2 )
=> ~ ( ord_less_eq_set_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_793_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_794_inf__sup__ord_I3_J,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_795_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_796_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_797_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_798_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_799_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_800_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_801_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_802_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_803_less__supI2,axiom,
! [X2: nat,B: nat,A: nat] :
( ( ord_less_nat @ X2 @ B )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_804_less__supI1,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_nat @ X2 @ A )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_805_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_806_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( sup_sup_set_nat @ A5 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_807_subset__UnE,axiom,
! [C2: set_nat,A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
=> ~ ! [A7: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ A4 )
=> ! [B9: set_nat] :
( ( ord_less_eq_set_nat @ B9 @ B5 )
=> ( C2
!= ( sup_sup_set_nat @ A7 @ B9 ) ) ) ) ) ).
% subset_UnE
thf(fact_808_Un__absorb2,axiom,
! [B5: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( sup_sup_set_nat @ A4 @ B5 )
= A4 ) ) ).
% Un_absorb2
thf(fact_809_Un__absorb1,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( sup_sup_set_nat @ A4 @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_810_Un__upper2,axiom,
! [B5: set_nat,A4: set_nat] : ( ord_less_eq_set_nat @ B5 @ ( sup_sup_set_nat @ A4 @ B5 ) ) ).
% Un_upper2
thf(fact_811_Un__upper1,axiom,
! [A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ A4 @ ( sup_sup_set_nat @ A4 @ B5 ) ) ).
% Un_upper1
thf(fact_812_Un__least,axiom,
! [A4: set_nat,C2: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B5 ) @ C2 ) ) ) ).
% Un_least
thf(fact_813_Un__mono,axiom,
! [A4: set_nat,C2: set_nat,B5: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B5 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B5 ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_814_Un__empty__left,axiom,
! [B5: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_815_Un__empty__right,axiom,
! [A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ bot_bot_set_nat )
= A4 ) ).
% Un_empty_right
thf(fact_816_UnI2,axiom,
! [C: a,B5: set_a,A4: set_a] :
( ( member_a @ C @ B5 )
=> ( member_a @ C @ ( sup_sup_set_a @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_817_UnI2,axiom,
! [C: nat,B5: set_nat,A4: set_nat] :
( ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_818_UnI1,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ A4 )
=> ( member_a @ C @ ( sup_sup_set_a @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_819_UnI1,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_820_UnE,axiom,
! [C: a,A4: set_a,B5: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A4 @ B5 ) )
=> ( ~ ( member_a @ C @ A4 )
=> ( member_a @ C @ B5 ) ) ) ).
% UnE
thf(fact_821_UnE,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) )
=> ( ~ ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% UnE
thf(fact_822_distrib__sup__le,axiom,
! [X2: nat,Y2: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( sup_sup_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_823_distrib__sup__le,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z2 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_824_distrib__inf__le,axiom,
! [X2: nat,Y2: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( inf_inf_nat @ X2 @ Z2 ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_825_distrib__inf__le,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat @ X2 @ Z2 ) ) @ ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_826_ivl__disj__un__two__touch_I4_J,axiom,
! [L2: a,M4: a,U: a] :
( ( ord_less_eq_a @ L2 @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L2 @ M4 ) @ ( set_or672772299803893939Most_a @ M4 @ U ) )
= ( set_or672772299803893939Most_a @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_827_ivl__disj__un__two__touch_I4_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( ord_less_eq_nat @ L2 @ M4 )
=> ( ( ord_less_eq_nat @ M4 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ M4 ) @ ( set_or1269000886237332187st_nat @ M4 @ U ) )
= ( set_or1269000886237332187st_nat @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_828_insert__is__Un,axiom,
( insert_nat
= ( ^ [A2: nat] : ( sup_sup_set_nat @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_829_Un__singleton__iff,axiom,
! [A4: set_nat,B5: set_nat,X2: nat] :
( ( ( sup_sup_set_nat @ A4 @ B5 )
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
= ( ( ( A4 = bot_bot_set_nat )
& ( B5
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A4
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B5 = bot_bot_set_nat ) )
| ( ( A4
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B5
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_830_singleton__Un__iff,axiom,
! [X2: nat,A4: set_nat,B5: set_nat] :
( ( ( insert_nat @ X2 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A4 @ B5 ) )
= ( ( ( A4 = bot_bot_set_nat )
& ( B5
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A4
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B5 = bot_bot_set_nat ) )
| ( ( A4
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B5
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_831_Un__Int__assoc__eq,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ C2 )
= ( inf_inf_set_nat @ A4 @ ( sup_sup_set_nat @ B5 @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A4 ) ) ).
% Un_Int_assoc_eq
thf(fact_832_Diff__partition,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( sup_sup_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ A4 ) )
= B5 ) ) ).
% Diff_partition
thf(fact_833_Diff__subset__conv,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ C2 )
= ( ord_less_eq_set_nat @ A4 @ ( sup_sup_set_nat @ B5 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_834_ivl__disj__un__two_I6_J,axiom,
! [L2: a,M4: a,U: a] :
( ( ord_less_eq_a @ L2 @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L2 @ M4 ) @ ( set_or4472690218693186638Most_a @ M4 @ U ) )
= ( set_or4472690218693186638Most_a @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_835_ivl__disj__un__two_I6_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( ord_less_eq_nat @ L2 @ M4 )
=> ( ( ord_less_eq_nat @ M4 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L2 @ M4 ) @ ( set_or6659071591806873216st_nat @ M4 @ U ) )
= ( set_or6659071591806873216st_nat @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_836_ivl__disj__un__two_I8_J,axiom,
! [L2: a,M4: a,U: a] :
( ( ord_less_eq_a @ L2 @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L2 @ M4 ) @ ( set_or4472690218693186638Most_a @ M4 @ U ) )
= ( set_or672772299803893939Most_a @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_837_ivl__disj__un__two_I8_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( ord_less_eq_nat @ L2 @ M4 )
=> ( ( ord_less_eq_nat @ M4 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ M4 ) @ ( set_or6659071591806873216st_nat @ M4 @ U ) )
= ( set_or1269000886237332187st_nat @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_838_ivl__disj__un__one_I3_J,axiom,
! [L2: a,U: a] :
( ( ord_less_eq_a @ L2 @ U )
=> ( ( sup_sup_set_a @ ( set_ord_atMost_a @ L2 ) @ ( set_or4472690218693186638Most_a @ L2 @ U ) )
= ( set_ord_atMost_a @ U ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_839_ivl__disj__un__one_I3_J,axiom,
! [L2: nat,U: nat] :
( ( ord_less_eq_nat @ L2 @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_atMost_nat @ L2 ) @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
= ( set_ord_atMost_nat @ U ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_840_ivl__disj__un__two__touch_I3_J,axiom,
! [L2: a,M4: a,U: a] :
( ( ord_less_a @ L2 @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L2 @ M4 ) @ ( set_or672772299803893939Most_a @ M4 @ U ) )
= ( set_or4472690218693186638Most_a @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_841_ivl__disj__un__two__touch_I3_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( ord_less_nat @ L2 @ M4 )
=> ( ( ord_less_eq_nat @ M4 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L2 @ M4 ) @ ( set_or1269000886237332187st_nat @ M4 @ U ) )
= ( set_or6659071591806873216st_nat @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_842_ivl__disj__un__two_I2_J,axiom,
! [L2: a,M4: a,U: a] :
( ( ord_less_eq_a @ L2 @ M4 )
=> ( ( ord_less_a @ M4 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L2 @ M4 ) @ ( set_or5939364468397584554Than_a @ M4 @ U ) )
= ( set_or5939364468397584554Than_a @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_843_ivl__disj__un__two_I2_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( ord_less_eq_nat @ L2 @ M4 )
=> ( ( ord_less_nat @ M4 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L2 @ M4 ) @ ( set_or5834768355832116004an_nat @ M4 @ U ) )
= ( set_or5834768355832116004an_nat @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_844_ivl__disj__un__singleton_I3_J,axiom,
! [L2: nat,U: nat] :
( ( ord_less_nat @ L2 @ U )
=> ( ( sup_sup_set_nat @ ( insert_nat @ L2 @ bot_bot_set_nat ) @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
= ( set_or4665077453230672383an_nat @ L2 @ U ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_845_ivl__disj__un__two_I4_J,axiom,
! [L2: a,M4: a,U: a] :
( ( ord_less_eq_a @ L2 @ M4 )
=> ( ( ord_less_a @ M4 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L2 @ M4 ) @ ( set_or5939364468397584554Than_a @ M4 @ U ) )
= ( set_or5139330845457685135Than_a @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_846_ivl__disj__un__two_I4_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( ord_less_eq_nat @ L2 @ M4 )
=> ( ( ord_less_nat @ M4 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ M4 ) @ ( set_or5834768355832116004an_nat @ M4 @ U ) )
= ( set_or4665077453230672383an_nat @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_847_ivl__disj__un__singleton_I6_J,axiom,
! [L2: a,U: a] :
( ( ord_less_eq_a @ L2 @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L2 @ U ) @ ( insert_a @ U @ bot_bot_set_a ) )
= ( set_or672772299803893939Most_a @ L2 @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_848_ivl__disj__un__singleton_I6_J,axiom,
! [L2: nat,U: nat] :
( ( ord_less_eq_nat @ L2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
= ( set_or1269000886237332187st_nat @ L2 @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_849_atLeastLessThan__iff,axiom,
! [I2: a,L2: a,U: a] :
( ( member_a @ I2 @ ( set_or5139330845457685135Than_a @ L2 @ U ) )
= ( ( ord_less_eq_a @ L2 @ I2 )
& ( ord_less_a @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_850_atLeastLessThan__iff,axiom,
! [I2: set_nat,L2: set_nat,U: set_nat] :
( ( member_set_nat @ I2 @ ( set_or3540276404033026485et_nat @ L2 @ U ) )
= ( ( ord_less_eq_set_nat @ L2 @ I2 )
& ( ord_less_set_nat @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_851_atLeastLessThan__iff,axiom,
! [I2: nat,L2: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
= ( ( ord_less_eq_nat @ L2 @ I2 )
& ( ord_less_nat @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_852_ivl__subset,axiom,
! [I2: a,J: a,M4: a,N2: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ I2 @ J ) @ ( set_or5139330845457685135Than_a @ M4 @ N2 ) )
= ( ( ord_less_eq_a @ J @ I2 )
| ( ( ord_less_eq_a @ M4 @ I2 )
& ( ord_less_eq_a @ J @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_853_ivl__subset,axiom,
! [I2: nat,J: nat,M4: nat,N2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ M4 @ N2 ) )
= ( ( ord_less_eq_nat @ J @ I2 )
| ( ( ord_less_eq_nat @ M4 @ I2 )
& ( ord_less_eq_nat @ J @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_854_atLeastLessThan__empty,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( set_or5139330845457685135Than_a @ A @ B )
= bot_bot_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_855_atLeastLessThan__empty,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( set_or3540276404033026485et_nat @ A @ B )
= bot_bot_set_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_856_atLeastLessThan__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( set_or4665077453230672383an_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_857_atLeastLessThan__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or4665077453230672383an_nat @ A @ B ) )
= ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_858_atLeastLessThan__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_859_ivl__diff,axiom,
! [I2: a,N2: a,M4: a] :
( ( ord_less_eq_a @ I2 @ N2 )
=> ( ( minus_minus_set_a @ ( set_or5139330845457685135Than_a @ I2 @ M4 ) @ ( set_or5139330845457685135Than_a @ I2 @ N2 ) )
= ( set_or5139330845457685135Than_a @ N2 @ M4 ) ) ) ).
% ivl_diff
thf(fact_860_ivl__diff,axiom,
! [I2: nat,N2: nat,M4: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ M4 ) @ ( set_or4665077453230672383an_nat @ I2 @ N2 ) )
= ( set_or4665077453230672383an_nat @ N2 @ M4 ) ) ) ).
% ivl_diff
thf(fact_861_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( B = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_862_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_863_Ico__eq__Ico,axiom,
! [L2: nat,H: nat,L: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L2 @ H )
= ( set_or4665077453230672383an_nat @ L @ H2 ) )
= ( ( ( L2 = L )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L2 @ H )
& ~ ( ord_less_nat @ L @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_864_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( A = C )
& ( B = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_865_atLeastLessThan__subset__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ A @ B ) @ ( set_or5139330845457685135Than_a @ C @ D2 ) )
=> ( ( ord_less_eq_a @ B @ A )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_866_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_867_ivl__disj__un__two_I3_J,axiom,
! [L2: a,M4: a,U: a] :
( ( ord_less_eq_a @ L2 @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L2 @ M4 ) @ ( set_or5139330845457685135Than_a @ M4 @ U ) )
= ( set_or5139330845457685135Than_a @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_868_ivl__disj__un__two_I3_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( ord_less_eq_nat @ L2 @ M4 )
=> ( ( ord_less_eq_nat @ M4 @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L2 @ M4 ) @ ( set_or4665077453230672383an_nat @ M4 @ U ) )
= ( set_or4665077453230672383an_nat @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_869_ivl__disj__int__two_I3_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L2 @ M4 ) @ ( set_or4665077453230672383an_nat @ M4 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(3)
thf(fact_870_Fpow__mono,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A4 ) @ ( finite_Fpow_nat @ B5 ) ) ) ).
% Fpow_mono
thf(fact_871_empty__in__Fpow,axiom,
! [A4: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A4 ) ) ).
% empty_in_Fpow
thf(fact_872_ivl__disj__un__two_I7_J,axiom,
! [L2: a,M4: a,U: a] :
( ( ord_less_eq_a @ L2 @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L2 @ M4 ) @ ( set_or672772299803893939Most_a @ M4 @ U ) )
= ( set_or672772299803893939Most_a @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_873_ivl__disj__un__two_I7_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( ord_less_eq_nat @ L2 @ M4 )
=> ( ( ord_less_eq_nat @ M4 @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L2 @ M4 ) @ ( set_or1269000886237332187st_nat @ M4 @ U ) )
= ( set_or1269000886237332187st_nat @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_874_ivl__disj__int__two_I7_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L2 @ M4 ) @ ( set_or1269000886237332187st_nat @ M4 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(7)
thf(fact_875_ivl__disj__int__two_I1_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or5834768355832116004an_nat @ L2 @ M4 ) @ ( set_or4665077453230672383an_nat @ M4 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(1)
thf(fact_876_ivl__disj__un__one_I8_J,axiom,
! [L2: a,U: a] :
( ( ord_less_eq_a @ L2 @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L2 @ U ) @ ( set_ord_atLeast_a @ U ) )
= ( set_ord_atLeast_a @ L2 ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_877_ivl__disj__un__one_I8_J,axiom,
! [L2: nat,U: nat] :
( ( ord_less_eq_nat @ L2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) @ ( set_ord_atLeast_nat @ U ) )
= ( set_ord_atLeast_nat @ L2 ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_878_ivl__disj__int__one_I8_J,axiom,
! [L2: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) @ ( set_ord_atLeast_nat @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(8)
thf(fact_879_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A @ B ) @ ( set_or5139330845457685135Than_a @ C @ D2 ) )
= ( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ A )
& ( ord_less_a @ B @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_880_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or3540276404033026485et_nat @ C @ D2 ) )
= ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_set_nat @ B @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_881_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_nat @ B @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_882_ivl__disj__un__two__touch_I2_J,axiom,
! [L2: a,M4: a,U: a] :
( ( ord_less_eq_a @ L2 @ M4 )
=> ( ( ord_less_a @ M4 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L2 @ M4 ) @ ( set_or5139330845457685135Than_a @ M4 @ U ) )
= ( set_or5139330845457685135Than_a @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_883_ivl__disj__un__two__touch_I2_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( ord_less_eq_nat @ L2 @ M4 )
=> ( ( ord_less_nat @ M4 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ M4 ) @ ( set_or4665077453230672383an_nat @ M4 @ U ) )
= ( set_or4665077453230672383an_nat @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_884_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or4665077453230672383an_nat
= ( ^ [A2: nat,B2: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_885_ivl__disj__un__two_I1_J,axiom,
! [L2: a,M4: a,U: a] :
( ( ord_less_a @ L2 @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L2 @ M4 ) @ ( set_or5139330845457685135Than_a @ M4 @ U ) )
= ( set_or5939364468397584554Than_a @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_886_ivl__disj__un__two_I1_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( ord_less_nat @ L2 @ M4 )
=> ( ( ord_less_eq_nat @ M4 @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L2 @ M4 ) @ ( set_or4665077453230672383an_nat @ M4 @ U ) )
= ( set_or5834768355832116004an_nat @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_887_ivl__disj__un__two__touch_I1_J,axiom,
! [L2: nat,M4: nat,U: nat] :
( ( ord_less_nat @ L2 @ M4 )
=> ( ( ord_less_nat @ M4 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L2 @ M4 ) @ ( set_or4665077453230672383an_nat @ M4 @ U ) )
= ( set_or5834768355832116004an_nat @ L2 @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_888_finite__induct__select,axiom,
! [S2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T3: set_nat] :
( ( ord_less_set_nat @ T3 @ S2 )
=> ( ( P @ T3 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ ( minus_minus_set_nat @ S2 @ T3 ) )
& ( P @ ( insert_nat @ X5 @ T3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_889_finite__remove__induct,axiom,
! [B5: set_a,P: set_a > $o] :
( ( finite_finite_a @ B5 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B5 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_890_finite__remove__induct,axiom,
! [B5: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B5 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B5 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_891_remove__induct,axiom,
! [P: set_a > $o,B5: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B5 )
=> ( P @ B5 ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B5 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% remove_induct
thf(fact_892_remove__induct,axiom,
! [P: set_nat > $o,B5: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B5 )
=> ( P @ B5 ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B5 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% remove_induct
thf(fact_893_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_894_finite__has__minimal2,axiom,
! [A4: set_a,A: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ A @ A4 )
=> ? [X3: a] :
( ( member_a @ X3 @ A4 )
& ( ord_less_eq_a @ X3 @ A )
& ! [Xa: a] :
( ( member_a @ Xa @ A4 )
=> ( ( ord_less_eq_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_895_finite__has__minimal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_896_finite__has__minimal2,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ( ord_less_eq_set_nat @ X3 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_897_finite__has__maximal2,axiom,
! [A4: set_a,A: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ A @ A4 )
=> ? [X3: a] :
( ( member_a @ X3 @ A4 )
& ( ord_less_eq_a @ A @ X3 )
& ! [Xa: a] :
( ( member_a @ Xa @ A4 )
=> ( ( ord_less_eq_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_898_finite__has__maximal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_899_finite__has__maximal2,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ( ord_less_eq_set_nat @ A @ X3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_900_rev__finite__subset,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_901_infinite__super,axiom,
! [S2: set_nat,T4: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T4 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T4 ) ) ) ).
% infinite_super
thf(fact_902_finite__subset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_903_infinite__Ici,axiom,
! [A: nat] :
~ ( finite_finite_nat @ ( set_ord_atLeast_nat @ A ) ) ).
% infinite_Ici
thf(fact_904_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_905_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_906_finite__psubset__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [B10: set_nat] :
( ( ord_less_set_nat @ B10 @ A8 )
=> ( P @ B10 ) )
=> ( P @ A8 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_907_finite__has__maximal,axiom,
! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ? [X3: a] :
( ( member_a @ X3 @ A4 )
& ! [Xa: a] :
( ( member_a @ Xa @ A4 )
=> ( ( ord_less_eq_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_908_finite__has__maximal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_909_finite__has__maximal,axiom,
! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_910_finite__has__minimal,axiom,
! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ? [X3: a] :
( ( member_a @ X3 @ A4 )
& ! [Xa: a] :
( ( member_a @ Xa @ A4 )
=> ( ( ord_less_eq_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_911_finite__has__minimal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_912_finite__has__minimal,axiom,
! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_913_finite__surj,axiom,
! [A4: set_nat,B5: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) )
=> ( finite_finite_nat @ B5 ) ) ) ).
% finite_surj
thf(fact_914_finite__subset__image,axiom,
! [B5: set_nat,F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A4 )
& ( finite_finite_nat @ C5 )
& ( B5
= ( image_nat_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_915_ex__finite__subset__image,axiom,
! [F: nat > nat,A4: set_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A4 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 )
& ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_916_all__finite__subset__image,axiom,
! [F: nat > nat,A4: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A4 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 ) )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_917_infinite__finite__induct,axiom,
! [P: set_a > $o,A4: set_a] :
( ! [A8: set_a] :
( ~ ( finite_finite_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_918_infinite__finite__induct,axiom,
! [P: set_nat > $o,A4: set_nat] :
( ! [A8: set_nat] :
( ~ ( finite_finite_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_919_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_920_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_921_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_922_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_923_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A2: set_nat] :
( ( A2 = bot_bot_set_nat )
| ? [A5: set_nat,B2: nat] :
( ( A2
= ( insert_nat @ B2 @ A5 ) )
& ( finite_finite_nat @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_924_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A8: set_nat] :
( ? [A3: nat] :
( A
= ( insert_nat @ A3 @ A8 ) )
=> ~ ( finite_finite_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_925_finite__subset__induct_H,axiom,
! [F2: set_a,A4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A4 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A4 )
=> ( ( ord_less_eq_set_a @ F3 @ A4 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_926_finite__subset__induct_H,axiom,
! [F2: set_nat,A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A4 )
=> ( ( ord_less_eq_set_nat @ F3 @ A4 )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_927_finite__subset__induct,axiom,
! [F2: set_a,A4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A4 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A4 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_928_finite__subset__induct,axiom,
! [F2: set_nat,A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A4 )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_929_infinite__remove,axiom,
! [S2: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_930_infinite__coinduct,axiom,
! [X7: set_nat > $o,A4: set_nat] :
( ( X7 @ A4 )
=> ( ! [A8: set_nat] :
( ( X7 @ A8 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A8 )
& ( ( X7 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_931_finite__empty__induct,axiom,
! [A4: set_a,P: set_a > $o] :
( ( finite_finite_a @ A4 )
=> ( ( P @ A4 )
=> ( ! [A3: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( member_a @ A3 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_932_finite__empty__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P @ A4 )
=> ( ! [A3: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( member_nat @ A3 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_933_finite__linorder__max__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( ord_less_nat @ X5 @ B3 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B3 @ A8 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_934_finite__linorder__min__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( ord_less_nat @ B3 @ X5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B3 @ A8 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_935_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > a] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S3 )
=> ( ord_less_eq_a @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_936_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > a] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S3 )
=> ( ord_less_eq_a @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_937_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_938_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_939_infinite__growing,axiom,
! [X7: set_a] :
( ( X7 != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ X7 )
=> ? [Xa: a] :
( ( member_a @ Xa @ X7 )
& ( ord_less_a @ X3 @ Xa ) ) )
=> ~ ( finite_finite_a @ X7 ) ) ) ).
% infinite_growing
thf(fact_940_infinite__growing,axiom,
! [X7: set_nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X7 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X7 ) ) ) ).
% infinite_growing
thf(fact_941_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_942_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ S2 )
& ( ord_less_nat @ ( F @ X5 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_943_arg__min__least,axiom,
! [S2: set_a,Y2: a,F: a > a] :
( ( finite_finite_a @ S2 )
=> ( ( S2 != bot_bot_set_a )
=> ( ( member_a @ Y2 @ S2 )
=> ( ord_less_eq_a @ ( F @ ( lattic3288624042836100505on_a_a @ F @ S2 ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_944_arg__min__least,axiom,
! [S2: set_nat,Y2: nat,F: nat > a] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y2 @ S2 )
=> ( ord_less_eq_a @ ( F @ ( lattic1148846883994911187_nat_a @ F @ S2 ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_945_arg__min__least,axiom,
! [S2: set_a,Y2: a,F: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( S2 != bot_bot_set_a )
=> ( ( member_a @ Y2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S2 ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_946_arg__min__least,axiom,
! [S2: set_nat,Y2: nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_947_finite__transitivity__chain,axiom,
! [A4: set_a,R: a > a > $o] :
( ( finite_finite_a @ A4 )
=> ( ! [X3: a] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: a,Y4: a,Z4: a] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z4 )
=> ( R @ X3 @ Z4 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ? [Y5: a] :
( ( member_a @ Y5 @ A4 )
& ( R @ X3 @ Y5 ) ) )
=> ( A4 = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_948_finite__transitivity__chain,axiom,
! [A4: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [X3: nat] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z4 )
=> ( R @ X3 @ Z4 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ A4 )
& ( R @ X3 @ Y5 ) ) )
=> ( A4 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_949_finite__indexed__bound,axiom,
! [A4: set_a,P: a > a > $o] :
( ( finite_finite_a @ A4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ? [X_1: a] : ( P @ X3 @ X_1 ) )
=> ? [M5: a] :
! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ? [K2: a] :
( ( ord_less_eq_a @ K2 @ M5 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_950_finite__indexed__bound,axiom,
! [A4: set_nat,P: nat > a > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ? [X_1: a] : ( P @ X3 @ X_1 ) )
=> ? [M5: a] :
! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ? [K2: a] :
( ( ord_less_eq_a @ K2 @ M5 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_951_finite__indexed__bound,axiom,
! [A4: set_a,P: a > nat > $o] :
( ( finite_finite_a @ A4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M5: nat] :
! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M5 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_952_finite__indexed__bound,axiom,
! [A4: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M5: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M5 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_953_Sup__fin_Oinsert__remove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A4 ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_954_Sup__fin_Oremove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A4 )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A4 )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_955_Sup__fin_Osingleton,axiom,
! [X2: nat] :
( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Sup_fin.singleton
thf(fact_956_inf__Sup__absorb,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ( inf_inf_nat @ A @ ( lattic1093996805478795353in_nat @ A4 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_957_Sup__fin_Oinsert,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_958_Sup__fin_OcoboundedI,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_959_Sup__fin_OcoboundedI,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ( ord_less_eq_set_nat @ A @ ( lattic3835124923745554447et_nat @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_960_Sup__fin_Obounded__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_961_Sup__fin_Obounded__iff,axiom,
! [A4: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A4 ) @ X2 )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ X @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_962_Sup__fin_OboundedI,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_nat @ A3 @ X2 ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_963_Sup__fin_OboundedI,axiom,
! [A4: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ A4 )
=> ( ord_less_eq_set_nat @ A3 @ X2 ) )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A4 ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_964_Sup__fin_OboundedE,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X2 )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A4 )
=> ( ord_less_eq_nat @ A9 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_965_Sup__fin_OboundedE,axiom,
! [A4: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A4 ) @ X2 )
=> ! [A9: set_nat] :
( ( member_set_nat @ A9 @ A4 )
=> ( ord_less_eq_set_nat @ A9 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_966_Sup__fin_Osubset__imp,axiom,
! [A4: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B5 )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A4 ) @ ( lattic3835124923745554447et_nat @ B5 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_967_Sup__fin_Osubset__imp,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B5 )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ B5 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_968_Sup__fin_Ohom__commute,axiom,
! [H: nat > nat,N3: set_nat] :
( ! [X3: nat,Y4: nat] :
( ( H @ ( sup_sup_nat @ X3 @ Y4 ) )
= ( sup_sup_nat @ ( H @ X3 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H @ ( lattic1093996805478795353in_nat @ N3 ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N3 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_969_Sup__fin_Osubset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B5 ) @ ( lattic1093996805478795353in_nat @ A4 ) )
= ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_970_Sup__fin_Oinsert__not__elem,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X2 @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_971_Sup__fin_Oclosed,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [X3: nat,Y4: nat] : ( member_nat @ ( sup_sup_nat @ X3 @ Y4 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ A4 ) ) ) ) ).
% Sup_fin.closed
thf(fact_972_Sup__fin_Ounion,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B5 )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ B5 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_973_Inf__fin_Oremove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A4 )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A4 )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_974_Inf__fin_Oinsert__remove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A4 ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_975_Inf__fin_Oinsert,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_976_Inf__fin_Osingleton,axiom,
! [X2: nat] :
( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Inf_fin.singleton
thf(fact_977_Inf__fin_OcoboundedI,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_978_Inf__fin_OcoboundedI,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A4 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_979_Inf__fin_Oin__idem,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A4 ) )
= ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_980_Inf__fin_OboundedE,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A4 ) )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A4 )
=> ( ord_less_eq_nat @ X2 @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_981_Inf__fin_OboundedE,axiom,
! [A4: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ A4 ) )
=> ! [A9: set_nat] :
( ( member_set_nat @ A9 @ A4 )
=> ( ord_less_eq_set_nat @ X2 @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_982_Inf__fin_OboundedI,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_nat @ X2 @ A3 ) )
=> ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_983_Inf__fin_OboundedI,axiom,
! [A4: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ A4 )
=> ( ord_less_eq_set_nat @ X2 @ A3 ) )
=> ( ord_less_eq_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ A4 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_984_Inf__fin_Obounded__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A4 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_985_Inf__fin_Obounded__iff,axiom,
! [A4: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ A4 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ X2 @ X ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_986_Inf__fin_Osubset__imp,axiom,
! [A4: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B5 )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B5 ) @ ( lattic3014633134055518761et_nat @ A4 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_987_Inf__fin_Osubset__imp,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B5 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B5 ) @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_988_Inf__fin_Ohom__commute,axiom,
! [H: nat > nat,N3: set_nat] :
( ! [X3: nat,Y4: nat] :
( ( H @ ( inf_inf_nat @ X3 @ Y4 ) )
= ( inf_inf_nat @ ( H @ X3 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H @ ( lattic5238388535129920115in_nat @ N3 ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H @ N3 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_989_Inf__fin_Osubset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B5 ) @ ( lattic5238388535129920115in_nat @ A4 ) )
= ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_990_Inf__fin_Oclosed,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [X3: nat,Y4: nat] : ( member_nat @ ( inf_inf_nat @ X3 @ Y4 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ A4 ) ) ) ) ).
% Inf_fin.closed
thf(fact_991_Inf__fin_Oinsert__not__elem,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X2 @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_992_Inf__fin_Ounion,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B5 )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ ( lattic5238388535129920115in_nat @ B5 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_993_Inf__fin__le__Sup__fin,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_994_Inf__fin__le__Sup__fin,axiom,
! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A4 ) @ ( lattic3835124923745554447et_nat @ A4 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_995_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic6009151579333465974et_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_996_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic3109210760196336428et_nat @ inf_inf_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_997_inf__img__fin__dom_H,axiom,
! [F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_998_inf__img__fin__domE_H,axiom,
! [F: nat > a,A4: set_nat] :
( ( finite_finite_a @ ( image_nat_a @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ~ ! [Y4: a] :
( ( member_a @ Y4 @ ( image_nat_a @ F @ A4 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_a @ F @ ( insert_a @ Y4 @ bot_bot_set_a ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_999_inf__img__fin__domE_H,axiom,
! [F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ~ ! [Y4: nat] :
( ( member_nat @ Y4 @ ( image_nat_nat @ F @ A4 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_1000_vimageI,axiom,
! [F: a > a,A: a,B: a,B5: set_a] :
( ( ( F @ A )
= B )
=> ( ( member_a @ B @ B5 )
=> ( member_a @ A @ ( vimage_a_a @ F @ B5 ) ) ) ) ).
% vimageI
thf(fact_1001_vimageI,axiom,
! [F: nat > a,A: nat,B: a,B5: set_a] :
( ( ( F @ A )
= B )
=> ( ( member_a @ B @ B5 )
=> ( member_nat @ A @ ( vimage_nat_a @ F @ B5 ) ) ) ) ).
% vimageI
thf(fact_1002_vimageI,axiom,
! [F: a > nat,A: a,B: nat,B5: set_nat] :
( ( ( F @ A )
= B )
=> ( ( member_nat @ B @ B5 )
=> ( member_a @ A @ ( vimage_a_nat @ F @ B5 ) ) ) ) ).
% vimageI
thf(fact_1003_vimageI,axiom,
! [F: nat > nat,A: nat,B: nat,B5: set_nat] :
( ( ( F @ A )
= B )
=> ( ( member_nat @ B @ B5 )
=> ( member_nat @ A @ ( vimage_nat_nat @ F @ B5 ) ) ) ) ).
% vimageI
thf(fact_1004_vimage__eq,axiom,
! [A: a,F: a > a,B5: set_a] :
( ( member_a @ A @ ( vimage_a_a @ F @ B5 ) )
= ( member_a @ ( F @ A ) @ B5 ) ) ).
% vimage_eq
thf(fact_1005_vimage__eq,axiom,
! [A: a,F: a > nat,B5: set_nat] :
( ( member_a @ A @ ( vimage_a_nat @ F @ B5 ) )
= ( member_nat @ ( F @ A ) @ B5 ) ) ).
% vimage_eq
thf(fact_1006_vimage__eq,axiom,
! [A: nat,F: nat > a,B5: set_a] :
( ( member_nat @ A @ ( vimage_nat_a @ F @ B5 ) )
= ( member_a @ ( F @ A ) @ B5 ) ) ).
% vimage_eq
thf(fact_1007_vimage__eq,axiom,
! [A: nat,F: nat > nat,B5: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F @ B5 ) )
= ( member_nat @ ( F @ A ) @ B5 ) ) ).
% vimage_eq
thf(fact_1008_vimage__empty,axiom,
! [F: nat > nat] :
( ( vimage_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_1009_image__subset__iff__subset__vimage,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 )
= ( ord_less_eq_set_nat @ A4 @ ( vimage_nat_nat @ F @ B5 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1010_vimage__singleton__eq,axiom,
! [A: a,F: a > nat,B: nat] :
( ( member_a @ A @ ( vimage_a_nat @ F @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_1011_vimage__singleton__eq,axiom,
! [A: nat,F: nat > nat,B: nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_1012_vimageD,axiom,
! [A: a,F: a > a,A4: set_a] :
( ( member_a @ A @ ( vimage_a_a @ F @ A4 ) )
=> ( member_a @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_1013_vimageD,axiom,
! [A: a,F: a > nat,A4: set_nat] :
( ( member_a @ A @ ( vimage_a_nat @ F @ A4 ) )
=> ( member_nat @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_1014_vimageD,axiom,
! [A: nat,F: nat > a,A4: set_a] :
( ( member_nat @ A @ ( vimage_nat_a @ F @ A4 ) )
=> ( member_a @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_1015_vimageD,axiom,
! [A: nat,F: nat > nat,A4: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F @ A4 ) )
=> ( member_nat @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_1016_vimageE,axiom,
! [A: a,F: a > a,B5: set_a] :
( ( member_a @ A @ ( vimage_a_a @ F @ B5 ) )
=> ( member_a @ ( F @ A ) @ B5 ) ) ).
% vimageE
thf(fact_1017_vimageE,axiom,
! [A: a,F: a > nat,B5: set_nat] :
( ( member_a @ A @ ( vimage_a_nat @ F @ B5 ) )
=> ( member_nat @ ( F @ A ) @ B5 ) ) ).
% vimageE
thf(fact_1018_vimageE,axiom,
! [A: nat,F: nat > a,B5: set_a] :
( ( member_nat @ A @ ( vimage_nat_a @ F @ B5 ) )
=> ( member_a @ ( F @ A ) @ B5 ) ) ).
% vimageE
thf(fact_1019_vimageE,axiom,
! [A: nat,F: nat > nat,B5: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F @ B5 ) )
=> ( member_nat @ ( F @ A ) @ B5 ) ) ).
% vimageE
thf(fact_1020_vimageI2,axiom,
! [F: a > a,A: a,A4: set_a] :
( ( member_a @ ( F @ A ) @ A4 )
=> ( member_a @ A @ ( vimage_a_a @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_1021_vimageI2,axiom,
! [F: nat > a,A: nat,A4: set_a] :
( ( member_a @ ( F @ A ) @ A4 )
=> ( member_nat @ A @ ( vimage_nat_a @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_1022_vimageI2,axiom,
! [F: a > nat,A: a,A4: set_nat] :
( ( member_nat @ ( F @ A ) @ A4 )
=> ( member_a @ A @ ( vimage_a_nat @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_1023_vimageI2,axiom,
! [F: nat > nat,A: nat,A4: set_nat] :
( ( member_nat @ ( F @ A ) @ A4 )
=> ( member_nat @ A @ ( vimage_nat_nat @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_1024_subset__vimage__iff,axiom,
! [A4: set_nat,F: nat > a,B5: set_a] :
( ( ord_less_eq_set_nat @ A4 @ ( vimage_nat_a @ F @ B5 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_a @ ( F @ X ) @ B5 ) ) ) ) ).
% subset_vimage_iff
thf(fact_1025_subset__vimage__iff,axiom,
! [A4: set_nat,F: nat > nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( vimage_nat_nat @ F @ B5 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_nat @ ( F @ X ) @ B5 ) ) ) ) ).
% subset_vimage_iff
thf(fact_1026_vimage__mono,axiom,
! [A4: set_nat,B5: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ A4 ) @ ( vimage_nat_nat @ F @ B5 ) ) ) ).
% vimage_mono
thf(fact_1027_inf__img__fin__domE,axiom,
! [F: nat > a,A4: set_nat] :
( ( finite_finite_a @ ( image_nat_a @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ~ ! [Y4: a] :
( ( member_a @ Y4 @ ( image_nat_a @ F @ A4 ) )
=> ( finite_finite_nat @ ( vimage_nat_a @ F @ ( insert_a @ Y4 @ bot_bot_set_a ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_1028_inf__img__fin__domE,axiom,
! [F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ~ ! [Y4: nat] :
( ( member_nat @ Y4 @ ( image_nat_nat @ F @ A4 ) )
=> ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_1029_inf__img__fin__dom,axiom,
! [F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_1030_finite__finite__vimage__IntI,axiom,
! [F2: set_a,H: nat > a,A4: set_nat] :
( ( finite_finite_a @ F2 )
=> ( ! [Y4: a] :
( ( member_a @ Y4 @ F2 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_a @ H @ ( insert_a @ Y4 @ bot_bot_set_a ) ) @ A4 ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_a @ H @ F2 ) @ A4 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1031_finite__finite__vimage__IntI,axiom,
! [F2: set_nat,H: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ F2 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ A4 ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H @ F2 ) @ A4 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1032_semilattice__order__set_Osubset__imp,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A4: set_nat,B5: set_nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B5 )
=> ( Less_eq @ ( lattic7742739596368939638_F_nat @ F @ B5 ) @ ( lattic7742739596368939638_F_nat @ F @ A4 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_1033_finite__vimageD_H,axiom,
! [F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ F @ A4 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_vimageD'
thf(fact_1034_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_1035_UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_1036_atMost__UNIV__triv,axiom,
( ( set_or4236626031148496127et_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% atMost_UNIV_triv
thf(fact_1037_atLeast__empty__triv,axiom,
( ( set_or1731685050470061051et_nat @ bot_bot_set_nat )
= top_top_set_set_nat ) ).
% atLeast_empty_triv
thf(fact_1038_inf__top_Oright__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_1039_inf__top_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A @ B ) )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1040_inf__top_Oleft__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1041_inf__top_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1042_top__eq__inf__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X2 @ Y2 ) )
= ( ( X2 = top_top_set_nat )
& ( Y2 = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_1043_inf__eq__top__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ( inf_inf_set_nat @ X2 @ Y2 )
= top_top_set_nat )
= ( ( X2 = top_top_set_nat )
& ( Y2 = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_1044_inf__top__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ top_top_set_nat )
= X2 ) ).
% inf_top_right
thf(fact_1045_inf__top__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_1046_Int__UNIV,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B5 )
= top_top_set_nat )
= ( ( A4 = top_top_set_nat )
& ( B5 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_1047_vimage__UNIV,axiom,
! [F: nat > nat] :
( ( vimage_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_1048_Diff__UNIV,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_1049_Un__UNIV__right,axiom,
! [A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_1050_Un__UNIV__left,axiom,
! [B5: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B5 )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_1051_subset__UNIV,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_1052_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_1053_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_1054_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_1055_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_1056_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_1057_boolean__algebra_Oconj__one__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ top_top_set_nat )
= X2 ) ).
% boolean_algebra.conj_one_right
thf(fact_1058_atMost__eq__UNIV__iff,axiom,
! [X2: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X2 )
= top_top_set_set_nat )
= ( X2 = top_top_set_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_1059_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_1060_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_1061_UNIV__eq__I,axiom,
! [A4: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A4 )
=> ( top_top_set_a = A4 ) ) ).
% UNIV_eq_I
thf(fact_1062_UNIV__eq__I,axiom,
! [A4: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A4 )
=> ( top_top_set_nat = A4 ) ) ).
% UNIV_eq_I
thf(fact_1063_not__UNIV__eq__Icc,axiom,
! [L: nat,H2: nat] :
( top_top_set_nat
!= ( set_or1269000886237332187st_nat @ L @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_1064_Int__UNIV__left,axiom,
! [B5: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B5 )
= B5 ) ).
% Int_UNIV_left
thf(fact_1065_Int__UNIV__right,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ top_top_set_nat )
= A4 ) ).
% Int_UNIV_right
thf(fact_1066_insert__UNIV,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_1067_not__UNIV__eq__Iic,axiom,
! [H2: nat] :
( top_top_set_nat
!= ( set_ord_atMost_nat @ H2 ) ) ).
% not_UNIV_eq_Iic
thf(fact_1068_range__eqI,axiom,
! [B: a,F: nat > a,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_nat_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1069_range__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1070_rangeI,axiom,
! [F: nat > a,X2: nat] : ( member_a @ ( F @ X2 ) @ ( image_nat_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1071_rangeI,axiom,
! [F: nat > nat,X2: nat] : ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1072_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_1073_atLeastAtMost__eq__UNIV__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ( set_or4548717258645045905et_nat @ X2 @ Y2 )
= top_top_set_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y2 = top_top_set_nat ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_1074_range__subsetD,axiom,
! [F: nat > a,B5: set_a,I2: nat] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ top_top_set_nat ) @ B5 )
=> ( member_a @ ( F @ I2 ) @ B5 ) ) ).
% range_subsetD
thf(fact_1075_range__subsetD,axiom,
! [F: nat > nat,B5: set_nat,I2: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B5 )
=> ( member_nat @ ( F @ I2 ) @ B5 ) ) ).
% range_subsetD
thf(fact_1076_not__UNIV__le__Icc,axiom,
! [L2: nat,H: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ H ) ) ).
% not_UNIV_le_Icc
thf(fact_1077_not__UNIV__le__Iic,axiom,
! [H: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_ord_atMost_nat @ H ) ) ).
% not_UNIV_le_Iic
thf(fact_1078_atLeast__eq__UNIV__iff,axiom,
! [X2: set_nat] :
( ( ( set_or1731685050470061051et_nat @ X2 )
= top_top_set_set_nat )
= ( X2 = bot_bot_set_nat ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_1079_atLeast__eq__UNIV__iff,axiom,
! [X2: nat] :
( ( ( set_ord_atLeast_nat @ X2 )
= top_top_set_nat )
= ( X2 = bot_bot_nat ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_1080_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_nat,X2: set_nat,Y2: set_nat] :
( ( ( inf_inf_set_nat @ A @ X2 )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A @ X2 )
= top_top_set_nat )
=> ( ( ( inf_inf_set_nat @ A @ Y2 )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A @ Y2 )
= top_top_set_nat )
=> ( X2 = Y2 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1081_range__eq__singletonD,axiom,
! [F: nat > nat,A: nat,X2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1082_semilattice__order__set_Obounded__iff,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A4: set_nat,X2: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( Less_eq @ X2 @ ( lattic7742739596368939638_F_nat @ F @ A4 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( Less_eq @ X2 @ X ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_1083_semilattice__order__set_OboundedI,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A4: set_a,X2: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A4 )
=> ( Less_eq @ X2 @ A3 ) )
=> ( Less_eq @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A4 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1084_semilattice__order__set_OboundedI,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A4: set_nat,X2: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( Less_eq @ X2 @ A3 ) )
=> ( Less_eq @ X2 @ ( lattic7742739596368939638_F_nat @ F @ A4 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1085_semilattice__order__set_OboundedE,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A4: set_a,X2: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( Less_eq @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A4 ) )
=> ! [A9: a] :
( ( member_a @ A9 @ A4 )
=> ( Less_eq @ X2 @ A9 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1086_semilattice__order__set_OboundedE,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A4: set_nat,X2: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( Less_eq @ X2 @ ( lattic7742739596368939638_F_nat @ F @ A4 ) )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A4 )
=> ( Less_eq @ X2 @ A9 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1087_surj__vimage__empty,axiom,
! [F: nat > nat,A4: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( vimage_nat_nat @ F @ A4 )
= bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_1088_vimage__subsetD,axiom,
! [F: nat > nat,B5: set_nat,A4: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ B5 ) @ A4 )
=> ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_1089_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_1090_semilattice__set_Oremove,axiom,
! [F: a > a > a,A4: set_a,X2: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A4 )
=> ( ( member_a @ X2 @ A4 )
=> ( ( ( ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) )
= bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ A4 )
= X2 ) )
& ( ( ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) )
!= bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ A4 )
= ( F @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_1091_semilattice__set_Oremove,axiom,
! [F: nat > nat > nat,A4: set_nat,X2: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ A4 )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ A4 )
= ( F @ X2 @ ( lattic7742739596368939638_F_nat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_1092_semilattice__set_Osingleton,axiom,
! [F: nat > nat > nat,X2: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ) ).
% semilattice_set.singleton
thf(fact_1093_semilattice__set_Ohom__commute,axiom,
! [F: nat > nat > nat,H: nat > nat,N3: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ! [X3: nat,Y4: nat] :
( ( H @ ( F @ X3 @ Y4 ) )
= ( F @ ( H @ X3 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H @ ( lattic7742739596368939638_F_nat @ F @ N3 ) )
= ( lattic7742739596368939638_F_nat @ F @ ( image_nat_nat @ H @ N3 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_1094_semilattice__set_Osubset,axiom,
! [F: nat > nat > nat,A4: set_nat,B5: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( F @ ( lattic7742739596368939638_F_nat @ F @ B5 ) @ ( lattic7742739596368939638_F_nat @ F @ A4 ) )
= ( lattic7742739596368939638_F_nat @ F @ A4 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_1095_semilattice__set_Oclosed,axiom,
! [F: a > a > a,A4: set_a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ! [X3: a,Y4: a] : ( member_a @ ( F @ X3 @ Y4 ) @ ( insert_a @ X3 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) )
=> ( member_a @ ( lattic5116578512385870296ce_F_a @ F @ A4 ) @ A4 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_1096_semilattice__set_Oclosed,axiom,
! [F: nat > nat > nat,A4: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [X3: nat,Y4: nat] : ( member_nat @ ( F @ X3 @ Y4 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic7742739596368939638_F_nat @ F @ A4 ) @ A4 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_1097_semilattice__set_Oinsert,axiom,
! [F: nat > nat > nat,A4: set_nat,X2: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X2 @ A4 ) )
= ( F @ X2 @ ( lattic7742739596368939638_F_nat @ F @ A4 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_1098_semilattice__set_Oinsert__not__elem,axiom,
! [F: a > a > a,A4: set_a,X2: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A4 )
=> ( ~ ( member_a @ X2 @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ ( insert_a @ X2 @ A4 ) )
= ( F @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A4 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_1099_semilattice__set_Oinsert__not__elem,axiom,
! [F: nat > nat > nat,A4: set_nat,X2: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X2 @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X2 @ A4 ) )
= ( F @ X2 @ ( lattic7742739596368939638_F_nat @ F @ A4 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_1100_semilattice__set_Ounion,axiom,
! [F: nat > nat > nat,A4: set_nat,B5: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B5 )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( F @ ( lattic7742739596368939638_F_nat @ F @ A4 ) @ ( lattic7742739596368939638_F_nat @ F @ B5 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_1101_semilattice__set_Oinsert__remove,axiom,
! [F: nat > nat > nat,A4: set_nat,X2: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X2 @ A4 ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X2 @ A4 ) )
= ( F @ X2 @ ( lattic7742739596368939638_F_nat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_1102_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila1667268886620078168et_nat @ inf_inf_set_nat @ top_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1103_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_1104_inj__on__insert,axiom,
! [F: nat > a,A: nat,A4: set_nat] :
( ( inj_on_nat_a @ F @ ( insert_nat @ A @ A4 ) )
= ( ( inj_on_nat_a @ F @ A4 )
& ~ ( member_a @ ( F @ A ) @ ( image_nat_a @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1105_inj__on__insert,axiom,
! [F: nat > nat,A: nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ ( insert_nat @ A @ A4 ) )
= ( ( inj_on_nat_nat @ F @ A4 )
& ~ ( member_nat @ ( F @ A ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1106_inj__on__image__mem__iff,axiom,
! [F: a > a,B5: set_a,A: a,A4: set_a] :
( ( inj_on_a_a @ F @ B5 )
=> ( ( member_a @ A @ B5 )
=> ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( member_a @ ( F @ A ) @ ( image_a_a @ F @ A4 ) )
= ( member_a @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1107_inj__on__image__mem__iff,axiom,
! [F: a > nat,B5: set_a,A: a,A4: set_a] :
( ( inj_on_a_nat @ F @ B5 )
=> ( ( member_a @ A @ B5 )
=> ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( member_nat @ ( F @ A ) @ ( image_a_nat @ F @ A4 ) )
= ( member_a @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1108_inj__on__image__mem__iff,axiom,
! [F: nat > a,B5: set_nat,A: nat,A4: set_nat] :
( ( inj_on_nat_a @ F @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_a @ ( F @ A ) @ ( image_nat_a @ F @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1109_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B5: set_nat,A: nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ ( F @ A ) @ ( image_nat_nat @ F @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1110_inj__image__subset__iff,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B5 ) )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_1111_finite__surj__inj,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ A4 @ ( image_nat_nat @ F @ A4 ) )
=> ( inj_on_nat_nat @ F @ A4 ) ) ) ).
% finite_surj_inj
thf(fact_1112_inj__on__finite,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_1113_endo__inj__surj,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ A4 )
=> ( ( inj_on_nat_nat @ F @ A4 )
=> ( ( image_nat_nat @ F @ A4 )
= A4 ) ) ) ) ).
% endo_inj_surj
thf(fact_1114_finite__vimage__IntI,axiom,
! [F2: set_nat,H: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( inj_on_nat_nat @ H @ A4 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H @ F2 ) @ A4 ) ) ) ) ).
% finite_vimage_IntI
thf(fact_1115_vimage__subsetI,axiom,
! [F: nat > nat,B5: set_nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ B5 ) @ A4 ) ) ) ).
% vimage_subsetI
thf(fact_1116_inj__on__iff__surj,axiom,
! [A4: set_nat,A10: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ A4 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A4 ) @ A10 ) ) )
= ( ? [G2: nat > nat] :
( ( image_nat_nat @ G2 @ A10 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1117_inj__on__image__subset__iff,axiom,
! [F: nat > nat,C2: set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ C2 )
=> ( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B5 ) )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_1118_infinite__countable__subset,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ? [F5: nat > nat] :
( ( inj_on_nat_nat @ F5 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ top_top_set_nat ) @ S2 ) ) ) ).
% infinite_countable_subset
thf(fact_1119_infinite__iff__countable__subset,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ top_top_set_nat ) @ S2 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_1120_subset__image__inj,axiom,
! [S2: set_nat,F: nat > nat,T4: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ ( image_nat_nat @ F @ T4 ) )
= ( ? [U3: set_nat] :
( ( ord_less_eq_set_nat @ U3 @ T4 )
& ( inj_on_nat_nat @ F @ U3 )
& ( S2
= ( image_nat_nat @ F @ U3 ) ) ) ) ) ).
% subset_image_inj
thf(fact_1121_ex__subset__image__inj,axiom,
! [F: nat > nat,S2: set_nat,P: set_nat > $o] :
( ( ? [T5: set_nat] :
( ( ord_less_eq_set_nat @ T5 @ ( image_nat_nat @ F @ S2 ) )
& ( P @ T5 ) ) )
= ( ? [T5: set_nat] :
( ( ord_less_eq_set_nat @ T5 @ S2 )
& ( inj_on_nat_nat @ F @ T5 )
& ( P @ ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_1122_all__subset__image__inj,axiom,
! [F: nat > nat,S2: set_nat,P: set_nat > $o] :
( ( ! [T5: set_nat] :
( ( ord_less_eq_set_nat @ T5 @ ( image_nat_nat @ F @ S2 ) )
=> ( P @ T5 ) ) )
= ( ! [T5: set_nat] :
( ( ( ord_less_eq_set_nat @ T5 @ S2 )
& ( inj_on_nat_nat @ F @ T5 ) )
=> ( P @ ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_1123_range__binary__eq,axiom,
! [A: nat,B: nat] :
( ( image_nat_nat @ ( sigma_binary_nat @ A @ B ) @ top_top_set_nat )
= ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ).
% range_binary_eq
thf(fact_1124_the__inv__into__into,axiom,
! [F: a > a,A4: set_a,X2: a,B5: set_a] :
( ( inj_on_a_a @ F @ A4 )
=> ( ( member_a @ X2 @ ( image_a_a @ F @ A4 ) )
=> ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( member_a @ ( the_inv_into_a_a @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_1125_the__inv__into__into,axiom,
! [F: a > nat,A4: set_a,X2: nat,B5: set_a] :
( ( inj_on_a_nat @ F @ A4 )
=> ( ( member_nat @ X2 @ ( image_a_nat @ F @ A4 ) )
=> ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( member_a @ ( the_inv_into_a_nat @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_1126_the__inv__into__into,axiom,
! [F: nat > a,A4: set_nat,X2: a,B5: set_nat] :
( ( inj_on_nat_a @ F @ A4 )
=> ( ( member_a @ X2 @ ( image_nat_a @ F @ A4 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( member_nat @ ( the_inv_into_nat_a @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_1127_the__inv__into__into,axiom,
! [F: nat > nat,A4: set_nat,X2: nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( member_nat @ X2 @ ( image_nat_nat @ F @ A4 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_1128_range__binaryset__eq,axiom,
! [A4: set_nat,B5: set_nat] :
( ( image_nat_set_nat @ ( sigma_binaryset_nat @ A4 @ B5 ) @ top_top_set_nat )
= ( insert_set_nat @ A4 @ ( insert_set_nat @ B5 @ ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ) ) ).
% range_binaryset_eq
thf(fact_1129_range__from__nat__into__subset,axiom,
! [A4: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ ( counta7321652538601044515to_nat @ A4 ) @ top_top_set_nat ) @ A4 ) ) ).
% range_from_nat_into_subset
thf(fact_1130_from__nat__into,axiom,
! [A4: set_a,N2: nat] :
( ( A4 != bot_bot_set_a )
=> ( member_a @ ( counta1652060073399151467into_a @ A4 @ N2 ) @ A4 ) ) ).
% from_nat_into
thf(fact_1131_from__nat__into,axiom,
! [A4: set_nat,N2: nat] :
( ( A4 != bot_bot_set_nat )
=> ( member_nat @ ( counta7321652538601044515to_nat @ A4 @ N2 ) @ A4 ) ) ).
% from_nat_into
thf(fact_1132_algebra__single__set,axiom,
! [X7: set_nat,S2: set_nat] :
( ( ord_less_eq_set_nat @ X7 @ S2 )
=> ( sigma_algebra_nat @ S2 @ ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ X7 @ ( insert_set_nat @ ( minus_minus_set_nat @ S2 @ X7 ) @ ( insert_set_nat @ S2 @ bot_bot_set_set_nat ) ) ) ) ) ) ).
% algebra_single_set
thf(fact_1133_sigma__algebra__single__set,axiom,
! [X7: set_nat,S2: set_nat] :
( ( ord_less_eq_set_nat @ X7 @ S2 )
=> ( sigma_8817008012692346403ra_nat @ S2 @ ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ X7 @ ( insert_set_nat @ ( minus_minus_set_nat @ S2 @ X7 ) @ ( insert_set_nat @ S2 @ bot_bot_set_set_nat ) ) ) ) ) ) ).
% sigma_algebra_single_set
thf(fact_1134_sigma__algebra__trivial,axiom,
! [Omega: set_nat] : ( sigma_8817008012692346403ra_nat @ Omega @ ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ Omega @ bot_bot_set_set_nat ) ) ) ).
% sigma_algebra_trivial
thf(fact_1135_closed__cdi__Un,axiom,
! [Omega: set_nat,M2: set_set_nat,A4: set_nat,B5: set_nat] :
( ( sigma_closed_cdi_nat @ Omega @ M2 )
=> ( ( member_set_nat @ bot_bot_set_nat @ M2 )
=> ( ( member_set_nat @ A4 @ M2 )
=> ( ( member_set_nat @ B5 @ M2 )
=> ( ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
=> ( member_set_nat @ ( sup_sup_set_nat @ A4 @ B5 ) @ M2 ) ) ) ) ) ) ).
% closed_cdi_Un
thf(fact_1136_partition__onD3,axiom,
! [A4: set_nat,P: set_set_nat] :
( ( disjoi4774308525696689793on_nat @ A4 @ P )
=> ~ ( member_set_nat @ bot_bot_set_nat @ P ) ) ).
% partition_onD3
thf(fact_1137_partition__on__empty,axiom,
! [P: set_set_nat] :
( ( disjoi4774308525696689793on_nat @ bot_bot_set_nat @ P )
= ( P = bot_bot_set_set_nat ) ) ).
% partition_on_empty
thf(fact_1138_partition__on__space,axiom,
! [A4: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( disjoi4774308525696689793on_nat @ A4 @ ( insert_set_nat @ A4 @ bot_bot_set_set_nat ) ) ) ).
% partition_on_space
thf(fact_1139_partition__on__restrict,axiom,
! [A4: set_nat,P: set_set_nat,B5: set_nat] :
( ( disjoi4774308525696689793on_nat @ A4 @ P )
=> ( disjoi4774308525696689793on_nat @ ( inf_inf_set_nat @ B5 @ A4 ) @ ( minus_2163939370556025621et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ B5 ) @ P ) @ ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ) ) ).
% partition_on_restrict
thf(fact_1140_card__vimage__inj,axiom,
! [F: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ A4 @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_nat @ F @ A4 ) )
= ( finite_card_nat @ A4 ) ) ) ) ).
% card_vimage_inj
thf(fact_1141_range__from__nat__into,axiom,
! [A4: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( counta1168086296615599829le_nat @ A4 )
=> ( ( image_nat_nat @ ( counta7321652538601044515to_nat @ A4 ) @ top_top_set_nat )
= A4 ) ) ) ).
% range_from_nat_into
thf(fact_1142_countable__empty,axiom,
counta1168086296615599829le_nat @ bot_bot_set_nat ).
% countable_empty
thf(fact_1143_from__nat__into__inject,axiom,
! [A4: set_nat,B5: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( counta1168086296615599829le_nat @ A4 )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( counta1168086296615599829le_nat @ B5 )
=> ( ( ( counta7321652538601044515to_nat @ A4 )
= ( counta7321652538601044515to_nat @ B5 ) )
= ( A4 = B5 ) ) ) ) ) ) ).
% from_nat_into_inject
thf(fact_1144_countable__Diff__eq,axiom,
! [A4: set_nat,X2: nat] :
( ( counta1168086296615599829le_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
= ( counta1168086296615599829le_nat @ A4 ) ) ).
% countable_Diff_eq
thf(fact_1145_card__less__sym__Diff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1146_pigeonhole,axiom,
! [F: nat > nat,A4: set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( finite_card_nat @ A4 ) )
=> ~ ( inj_on_nat_nat @ F @ A4 ) ) ).
% pigeonhole
thf(fact_1147_countable__subset__image,axiom,
! [B5: set_nat,F: nat > nat,A4: set_nat] :
( ( ( counta1168086296615599829le_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) ) )
= ( ? [A11: set_nat] :
( ( counta1168086296615599829le_nat @ A11 )
& ( ord_less_eq_set_nat @ A11 @ A4 )
& ( B5
= ( image_nat_nat @ F @ A11 ) ) ) ) ) ).
% countable_subset_image
thf(fact_1148_ex__countable__subset__image,axiom,
! [F: nat > nat,S2: set_nat,P: set_nat > $o] :
( ( ? [T5: set_nat] :
( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ ( image_nat_nat @ F @ S2 ) )
& ( P @ T5 ) ) )
= ( ? [T5: set_nat] :
( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S2 )
& ( P @ ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% ex_countable_subset_image
thf(fact_1149_all__countable__subset__image,axiom,
! [F: nat > nat,S2: set_nat,P: set_nat > $o] :
( ( ! [T5: set_nat] :
( ( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ ( image_nat_nat @ F @ S2 ) ) )
=> ( P @ T5 ) ) )
= ( ! [T5: set_nat] :
( ( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S2 ) )
=> ( P @ ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% all_countable_subset_image
thf(fact_1150_infinite__countable__subset_H,axiom,
! [X7: set_nat] :
( ~ ( finite_finite_nat @ X7 )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ X7 )
& ( counta1168086296615599829le_nat @ C5 )
& ~ ( finite_finite_nat @ C5 ) ) ) ).
% infinite_countable_subset'
thf(fact_1151_countable__subset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( counta1168086296615599829le_nat @ B5 )
=> ( counta1168086296615599829le_nat @ A4 ) ) ) ).
% countable_subset
thf(fact_1152_psubset__card__mono,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_set_nat @ A4 @ B5 )
=> ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ).
% psubset_card_mono
thf(fact_1153_card__le__if__inj__on__rel,axiom,
! [B5: set_a,A4: set_a,R2: a > a > $o] :
( ( finite_finite_a @ B5 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A4 )
=> ? [B4: a] :
( ( member_a @ B4 @ B5 )
& ( R2 @ A3 @ B4 ) ) )
=> ( ! [A1: a,A22: a,B3: a] :
( ( member_a @ A1 @ A4 )
=> ( ( member_a @ A22 @ A4 )
=> ( ( member_a @ B3 @ B5 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A4 ) @ ( finite_card_a @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1154_card__le__if__inj__on__rel,axiom,
! [B5: set_a,A4: set_nat,R2: nat > a > $o] :
( ( finite_finite_a @ B5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ? [B4: a] :
( ( member_a @ B4 @ B5 )
& ( R2 @ A3 @ B4 ) ) )
=> ( ! [A1: nat,A22: nat,B3: a] :
( ( member_nat @ A1 @ A4 )
=> ( ( member_nat @ A22 @ A4 )
=> ( ( member_a @ B3 @ B5 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_a @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1155_card__le__if__inj__on__rel,axiom,
! [B5: set_nat,A4: set_a,R2: a > nat > $o] :
( ( finite_finite_nat @ B5 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A4 )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B5 )
& ( R2 @ A3 @ B4 ) ) )
=> ( ! [A1: a,A22: a,B3: nat] :
( ( member_a @ A1 @ A4 )
=> ( ( member_a @ A22 @ A4 )
=> ( ( member_nat @ B3 @ B5 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1156_card__le__if__inj__on__rel,axiom,
! [B5: set_nat,A4: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B5 )
& ( R2 @ A3 @ B4 ) ) )
=> ( ! [A1: nat,A22: nat,B3: nat] :
( ( member_nat @ A1 @ A4 )
=> ( ( member_nat @ A22 @ A4 )
=> ( ( member_nat @ B3 @ B5 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1157_card__mono,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ).
% card_mono
thf(fact_1158_card__seteq,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B5 ) @ ( finite_card_nat @ A4 ) )
=> ( A4 = B5 ) ) ) ) ).
% card_seteq
thf(fact_1159_card__subset__eq,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ( finite_card_nat @ A4 )
= ( finite_card_nat @ B5 ) )
=> ( A4 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_1160_exists__subset__between,axiom,
! [A4: set_nat,N2: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C2 )
& ( ( finite_card_nat @ B8 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1161_obtain__subset__with__card__n,axiom,
! [N2: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S2 ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S2 )
=> ( ( ( finite_card_nat @ T3 )
= N2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1162_infinite__arbitrarily__large,axiom,
! [A4: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A4 )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N2 )
& ( ord_less_eq_set_nat @ B8 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1163_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F2 )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1164_all__countable__subset__image__inj,axiom,
! [F: nat > nat,S2: set_nat,P: set_nat > $o] :
( ( ! [T5: set_nat] :
( ( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ ( image_nat_nat @ F @ S2 ) ) )
=> ( P @ T5 ) ) )
= ( ! [T5: set_nat] :
( ( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S2 )
& ( inj_on_nat_nat @ F @ T5 ) )
=> ( P @ ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% all_countable_subset_image_inj
thf(fact_1165_ex__countable__subset__image__inj,axiom,
! [F: nat > nat,S2: set_nat,P: set_nat > $o] :
( ( ? [T5: set_nat] :
( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ ( image_nat_nat @ F @ S2 ) )
& ( P @ T5 ) ) )
= ( ? [T5: set_nat] :
( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S2 )
& ( inj_on_nat_nat @ F @ T5 )
& ( P @ ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% ex_countable_subset_image_inj
thf(fact_1166_uncountable__def,axiom,
! [A4: set_nat] :
( ( ~ ( counta1168086296615599829le_nat @ A4 ) )
= ( ( A4 != bot_bot_set_nat )
& ~ ? [F4: nat > nat] :
( ( image_nat_nat @ F4 @ top_top_set_nat )
= A4 ) ) ) ).
% uncountable_def
thf(fact_1167_surj__card__le,axiom,
! [A4: set_nat,B5: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B5 ) @ ( finite_card_nat @ A4 ) ) ) ) ).
% surj_card_le
thf(fact_1168_card__Diff__subset,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1169_card__Diff1__le,axiom,
! [A4: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ).
% card_Diff1_le
thf(fact_1170_card__Diff__subset__Int,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A4 @ B5 ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1171_sigma__algebra_Ocountable,axiom,
! [Omega: set_a,M2: set_set_a,A4: set_a] :
( ( sigma_4968961713055010667ebra_a @ Omega @ M2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A4 )
=> ( member_set_a @ ( insert_a @ A3 @ bot_bot_set_a ) @ M2 ) )
=> ( ( counta4098120917673242425able_a @ A4 )
=> ( member_set_a @ A4 @ M2 ) ) ) ) ).
% sigma_algebra.countable
thf(fact_1172_sigma__algebra_Ocountable,axiom,
! [Omega: set_nat,M2: set_set_nat,A4: set_nat] :
( ( sigma_8817008012692346403ra_nat @ Omega @ M2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( member_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) @ M2 ) )
=> ( ( counta1168086296615599829le_nat @ A4 )
=> ( member_set_nat @ A4 @ M2 ) ) ) ) ).
% sigma_algebra.countable
thf(fact_1173_card__psubset,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) )
=> ( ord_less_set_nat @ A4 @ B5 ) ) ) ) ).
% card_psubset
thf(fact_1174_countable__vimage,axiom,
! [B5: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ( counta1168086296615599829le_nat @ ( vimage_nat_nat @ F @ B5 ) )
=> ( counta1168086296615599829le_nat @ B5 ) ) ) ).
% countable_vimage
thf(fact_1175_surjective__iff__injective__gen,axiom,
! [S2: set_nat,T4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T4 )
=> ( ( ( finite_card_nat @ S2 )
= ( finite_card_nat @ T4 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ S2 ) @ T4 )
=> ( ( ! [X: nat] :
( ( member_nat @ X @ T4 )
=> ? [Y: nat] :
( ( member_nat @ Y @ S2 )
& ( ( F @ Y )
= X ) ) ) )
= ( inj_on_nat_nat @ F @ S2 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_1176_inj__on__iff__card__le,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ A4 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A4 ) @ B5 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_1177_card__inj__on__le,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_1178_card__le__inj,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) )
=> ? [F5: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ A4 ) @ B5 )
& ( inj_on_nat_nat @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_1179_card__bij__eq,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat,G4: nat > nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 )
=> ( ( inj_on_nat_nat @ G4 @ B5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G4 @ B5 ) @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( finite_card_nat @ A4 )
= ( finite_card_nat @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_1180_card__Diff1__less__iff,axiom,
! [A4: set_a,X2: a] :
( ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A4 ) )
= ( ( finite_finite_a @ A4 )
& ( member_a @ X2 @ A4 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1181_card__Diff1__less__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) )
= ( ( finite_finite_nat @ A4 )
& ( member_nat @ X2 @ A4 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1182_card__Diff2__less,axiom,
! [A4: set_a,X2: a,Y2: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ X2 @ A4 )
=> ( ( member_a @ Y2 @ A4 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ ( insert_a @ Y2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A4 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1183_card__Diff2__less,axiom,
! [A4: set_nat,X2: nat,Y2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( member_nat @ Y2 @ A4 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1184_card__Diff1__less,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ X2 @ A4 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A4 ) ) ) ) ).
% card_Diff1_less
thf(fact_1185_card__Diff1__less,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ) ) ).
% card_Diff1_less
thf(fact_1186_card__vimage__inj__on__le,axiom,
! [F: nat > nat,D: set_nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ D )
=> ( ( finite_finite_nat @ A4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ A4 ) @ D ) ) @ ( finite_card_nat @ A4 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1187_subset__range__from__nat__into,axiom,
! [A4: set_nat] :
( ( counta1168086296615599829le_nat @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ ( image_nat_nat @ ( counta7321652538601044515to_nat @ A4 ) @ top_top_set_nat ) ) ) ).
% subset_range_from_nat_into
thf(fact_1188_countable__separating__set__linorder2,axiom,
? [B8: set_nat] :
( ( counta1168086296615599829le_nat @ B8 )
& ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B8 )
& ( ord_less_eq_nat @ X5 @ Xa2 )
& ( ord_less_nat @ Xa2 @ Y5 ) ) ) ) ).
% countable_separating_set_linorder2
thf(fact_1189_countable__separating__set__linorder1,axiom,
? [B8: set_nat] :
( ( counta1168086296615599829le_nat @ B8 )
& ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B8 )
& ( ord_less_nat @ X5 @ Xa2 )
& ( ord_less_eq_nat @ Xa2 @ Y5 ) ) ) ) ).
% countable_separating_set_linorder1
thf(fact_1190_card__greaterThanAtMost,axiom,
! [L2: nat,U: nat] :
( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
= ( minus_minus_nat @ U @ L2 ) ) ).
% card_greaterThanAtMost
thf(fact_1191_card__atLeastLessThan,axiom,
! [L2: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
= ( minus_minus_nat @ U @ L2 ) ) ).
% card_atLeastLessThan
thf(fact_1192_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K @ M5 )
=> ? [N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ( member_nat @ N4 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_1193_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M3: nat] :
? [N: nat] :
( ( ord_less_nat @ M3 @ N )
& ( member_nat @ N @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1194_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1195_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_nat @ X @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1196_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_nat @ X3 @ N2 ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1197_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M2: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M2 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1198_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1199_less__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M4 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1200_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_1201_finite__greaterThanLessThan,axiom,
! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_1202_finite__atLeastAtMost,axiom,
! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).
% finite_atLeastAtMost
thf(fact_1203_finite__atLeastLessThan,axiom,
! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) ) ).
% finite_atLeastLessThan
thf(fact_1204_finite__greaterThanAtMost,axiom,
! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_1205_finite__nat__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [S4: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_1206_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_1207_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N6: nat] :
( ~ ( P @ N6 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N6 )
& ~ ( P @ M6 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_1208_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N6: nat] :
( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N6 )
=> ( P @ M6 ) )
=> ( P @ N6 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_1209_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1210_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_1211_less__not__refl2,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ N2 @ M4 )
=> ( M4 != N2 ) ) ).
% less_not_refl2
thf(fact_1212_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1213_nat__neq__iff,axiom,
! [M4: nat,N2: nat] :
( ( M4 != N2 )
= ( ( ord_less_nat @ M4 @ N2 )
| ( ord_less_nat @ N2 @ M4 ) ) ) ).
% nat_neq_iff
thf(fact_1214_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1215_le__neq__implies__less,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( M4 != N2 )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1216_less__or__eq__imp__le,axiom,
! [M4: nat,N2: nat] :
( ( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1217_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1218_less__imp__le__nat,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1219_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
& ( M3 != N ) ) ) ) ).
% nat_less_le
thf(fact_1220_diff__less__mono2,axiom,
! [M4: nat,N2: nat,L2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ( ord_less_nat @ M4 @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M4 ) ) ) ) ).
% diff_less_mono2
thf(fact_1221_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1222_ex__card,axiom,
! [N2: nat,A4: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ A4 ) )
=> ? [S3: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ A4 )
& ( ( finite_card_nat @ S3 )
= N2 ) ) ) ).
% ex_card
thf(fact_1223_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M4: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M4 ) ) ) ).
% nat_descend_induct
thf(fact_1224_finite__enumerate__mono__iff,axiom,
! [S2: set_nat,M4: nat,N2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ M4 @ ( finite_card_nat @ S2 ) )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S2 ) )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M4 ) @ ( infini8530281810654367211te_nat @ S2 @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_1225_finite__enum__subset,axiom,
! [X7: set_nat,Y7: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_nat @ X7 ) )
=> ( ( infini8530281810654367211te_nat @ X7 @ I3 )
= ( infini8530281810654367211te_nat @ Y7 @ I3 ) ) )
=> ( ( finite_finite_nat @ X7 )
=> ( ( finite_finite_nat @ Y7 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ X7 ) @ ( finite_card_nat @ Y7 ) )
=> ( ord_less_eq_set_nat @ X7 @ Y7 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_1226_enumerate__mono__le__iff,axiom,
! [S2: set_nat,M4: nat,N2: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S2 @ M4 ) @ ( infini8530281810654367211te_nat @ S2 @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% enumerate_mono_le_iff
thf(fact_1227_enumerate__mono__iff,axiom,
! [S2: set_nat,M4: nat,N2: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M4 ) @ ( infini8530281810654367211te_nat @ S2 @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ) ).
% enumerate_mono_iff
thf(fact_1228_enumerate__mono,axiom,
! [M4: nat,N2: nat,S2: set_nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M4 ) @ ( infini8530281810654367211te_nat @ S2 @ N2 ) ) ) ) ).
% enumerate_mono
thf(fact_1229_finite__enum__ext,axiom,
! [X7: set_nat,Y7: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_nat @ X7 ) )
=> ( ( infini8530281810654367211te_nat @ X7 @ I3 )
= ( infini8530281810654367211te_nat @ Y7 @ I3 ) ) )
=> ( ( finite_finite_nat @ X7 )
=> ( ( finite_finite_nat @ Y7 )
=> ( ( ( finite_card_nat @ X7 )
= ( finite_card_nat @ Y7 ) )
=> ( X7 = Y7 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_1230_finite__enumerate__Ex,axiom,
! [S2: set_nat,S: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( member_nat @ S @ S2 )
=> ? [N6: nat] :
( ( ord_less_nat @ N6 @ ( finite_card_nat @ S2 ) )
& ( ( infini8530281810654367211te_nat @ S2 @ N6 )
= S ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_1231_finite__enumerate__in__set,axiom,
! [S2: set_nat,N2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S2 ) )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S2 @ N2 ) @ S2 ) ) ) ).
% finite_enumerate_in_set
thf(fact_1232_finite__enumerate__mono,axiom,
! [M4: nat,N2: nat,S2: set_nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S2 ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M4 ) @ ( infini8530281810654367211te_nat @ S2 @ N2 ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_1233_finite__le__enumerate,axiom,
! [S2: set_nat,N2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S2 ) )
=> ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S2 @ N2 ) ) ) ) ).
% finite_le_enumerate
thf(fact_1234_card__range__greater__zero,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_1235_card__Diff__singleton,axiom,
! [X2: a,A4: set_a] :
( ( member_a @ X2 @ A4 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A4 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1236_card__Diff__singleton,axiom,
! [X2: nat,A4: set_nat] :
( ( member_nat @ X2 @ A4 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1237_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1238_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1239_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1240_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1241_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1242_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_1243_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_1244_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_1245_zero__less__diff,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M4 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% zero_less_diff
thf(fact_1246_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1247_card__0__eq,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( finite_card_nat @ A4 )
= zero_zero_nat )
= ( A4 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_1248_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1249_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1250_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1251_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1252_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1253_gr__implies__not0,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1254_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N6: nat] :
( ( ord_less_nat @ zero_zero_nat @ N6 )
=> ( ~ ( P @ N6 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N6 )
& ~ ( P @ M6 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1255_diff__less,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_nat @ ( minus_minus_nat @ M4 @ N2 ) @ M4 ) ) ) ).
% diff_less
thf(fact_1256_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1257_atLeastLessThan0,axiom,
! [M4: nat] :
( ( set_or4665077453230672383an_nat @ M4 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1258_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1259_gr__implies__not__zero,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1260_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1261_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_1262_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_1263_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_1264_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1265_card__insert__le__m1,axiom,
! [N2: nat,Y2: set_nat,X2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y2 ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X2 @ Y2 ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_1266_card__1__singletonE,axiom,
! [A4: set_nat] :
( ( ( finite_card_nat @ A4 )
= one_one_nat )
=> ~ ! [X3: nat] :
( A4
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_1267_card__eq__0__iff,axiom,
! [A4: set_nat] :
( ( ( finite_card_nat @ A4 )
= zero_zero_nat )
= ( ( A4 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A4 ) ) ) ).
% card_eq_0_iff
thf(fact_1268_card__ge__0__finite,axiom,
! [A4: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A4 ) )
=> ( finite_finite_nat @ A4 ) ) ).
% card_ge_0_finite
thf(fact_1269_subset__eq__atLeast0__atMost__finite,axiom,
! [N3: set_nat,N2: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
=> ( finite_finite_nat @ N3 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1270_subset__eq__atLeast0__lessThan__finite,axiom,
! [N3: set_nat,N2: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
=> ( finite_finite_nat @ N3 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_1271_subset__eq__atLeast0__lessThan__card,axiom,
! [N3: set_nat,N2: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N2 ) ) ).
% subset_eq_atLeast0_lessThan_card
% Conjectures (1)
thf(conj_0,conjecture,
member_a @ x @ i ).
%------------------------------------------------------------------------------