0.13/0.13 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.13/0.13 % Command : tptp2X_and_run_prover9 %d %s 0.13/0.35 % Computer : n013.cluster.edu 0.13/0.35 % Model : x86_64 x86_64 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.35 % Memory : 8042.1875MB 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.35 % CPULimit : 1200 0.13/0.35 % DateTime : Tue Jul 13 17:08:48 EDT 2021 0.13/0.35 % CPUTime : 1.03/1.31 ============================== Prover9 =============================== 1.03/1.31 Prover9 (32) version 2009-11A, November 2009. 1.03/1.31 Process 3915 was started by sandbox2 on n013.cluster.edu, 1.03/1.31 Tue Jul 13 17:08:49 2021 1.03/1.31 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 1200 -f /tmp/Prover9_3726_n013.cluster.edu". 1.03/1.31 ============================== end of head =========================== 1.03/1.31 1.03/1.31 ============================== INPUT ================================= 1.03/1.31 1.03/1.31 % Reading from file /tmp/Prover9_3726_n013.cluster.edu 1.03/1.31 1.03/1.31 set(prolog_style_variables). 1.03/1.31 set(auto2). 1.03/1.31 % set(auto2) -> set(auto). 1.03/1.31 % set(auto) -> set(auto_inference). 1.03/1.31 % set(auto) -> set(auto_setup). 1.03/1.31 % set(auto_setup) -> set(predicate_elim). 1.03/1.31 % set(auto_setup) -> assign(eq_defs, unfold). 1.03/1.31 % set(auto) -> set(auto_limits). 1.03/1.31 % set(auto_limits) -> assign(max_weight, "100.000"). 1.03/1.31 % set(auto_limits) -> assign(sos_limit, 20000). 1.03/1.31 % set(auto) -> set(auto_denials). 1.03/1.31 % set(auto) -> set(auto_process). 1.03/1.31 % set(auto2) -> assign(new_constants, 1). 1.03/1.31 % set(auto2) -> assign(fold_denial_max, 3). 1.03/1.31 % set(auto2) -> assign(max_weight, "200.000"). 1.03/1.31 % set(auto2) -> assign(max_hours, 1). 1.03/1.31 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.03/1.31 % set(auto2) -> assign(max_seconds, 0). 1.03/1.31 % set(auto2) -> assign(max_minutes, 5). 1.03/1.31 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.03/1.31 % set(auto2) -> set(sort_initial_sos). 1.03/1.31 % set(auto2) -> assign(sos_limit, -1). 1.03/1.31 % set(auto2) -> assign(lrs_ticks, 3000). 1.03/1.31 % set(auto2) -> assign(max_megs, 400). 1.03/1.31 % set(auto2) -> assign(stats, some). 1.03/1.31 % set(auto2) -> clear(echo_input). 1.03/1.31 % set(auto2) -> set(quiet). 1.03/1.31 % set(auto2) -> clear(print_initial_clauses). 1.03/1.31 % set(auto2) -> clear(print_given). 1.03/1.31 assign(lrs_ticks,-1). 1.03/1.31 assign(sos_limit,10000). 1.03/1.31 assign(order,kbo). 1.03/1.31 set(lex_order_vars). 1.03/1.31 clear(print_given). 1.03/1.31 1.03/1.31 % formulas(sos). % not echoed (339 formulas) 1.03/1.31 1.03/1.31 ============================== end of input ========================== 1.03/1.31 1.03/1.31 % From the command line: assign(max_seconds, 1200). 1.03/1.31 1.03/1.31 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.03/1.31 1.03/1.31 % Formulas that are not ordinary clauses: 1.03/1.31 1 (all A all B all C (relation(C) -> (in(A,relation_rng(relation_rng_restriction(B,C))) <-> in(A,B) & in(A,relation_rng(C))))) # label(t115_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 2 (all A all B all C (set_difference(A,B) = C <-> (all D (-in(D,B) & in(D,A) <-> in(D,C))))) # label(d4_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 3 $T # label(dt_k1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 4 (all A all B all C (subset(A,B) -> subset(A,set_difference(B,singleton(C))) | in(C,A))) # label(l3_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 5 (all A all B (element(B,powerset(A)) -> B = subset_complement(A,subset_complement(A,B)))) # label(involutiveness_k3_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 6 (all A all B (empty(A) & relation(B) -> empty(relation_composition(B,A)) & relation(relation_composition(B,A)))) # label(fc10_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 7 $T # label(dt_k4_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 8 (all A (relation(A) -> (reflexive(A) <-> (all B (in(B,relation_field(A)) -> in(ordered_pair(B,B),A)))))) # label(l1_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 9 (all A all B (relation(B) -> subset(relation_rng(relation_rng_restriction(A,B)),relation_rng(B)))) # label(t118_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 10 (all A (relation_dom(identity_relation(A)) = A & A = relation_rng(identity_relation(A)))) # label(t71_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 11 (all A all B (relation(B) -> subset(relation_dom(relation_rng_restriction(A,B)),relation_dom(B)))) # label(l29_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 12 (all A (relation(A) -> (all B (relation(B) -> (subset(A,B) -> subset(relation_rng(A),relation_rng(B)) & subset(relation_dom(A),relation_dom(B))))))) # label(t25_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 13 (all A (subset(A,empty_set) -> empty_set = A)) # label(t3_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 14 (all A (relation(A) -> (all B ((all C (in(C,B) -> in(ordered_pair(C,C),A))) <-> is_reflexive_in(A,B))))) # label(d1_relat_2) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 15 (all A all B (in(A,B) -> B = set_union2(singleton(A),B))) # label(t46_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 16 (all A all B (relation(B) -> ((all C all D (D = C & in(C,A) <-> in(ordered_pair(C,D),B))) <-> identity_relation(A) = B))) # label(d10_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 17 (all A all B (element(B,powerset(A)) -> (all C (in(C,B) -> in(C,A))))) # label(l3_subset_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 18 (all A all B all C (subset(A,B) & subset(B,C) -> subset(A,C))) # label(t1_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 19 (all A all B all C (relation(C) -> (in(A,relation_inverse_image(C,B)) <-> (exists D (in(D,relation_rng(C)) & in(D,B) & in(ordered_pair(A,D),C)))))) # label(t166_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 20 (all A (A != empty_set -> (all B (element(B,powerset(A)) -> (all C (element(C,A) -> (-in(C,B) -> in(C,subset_complement(A,B))))))))) # label(t50_subset_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 21 (exists A (relation(A) & empty(A))) # label(rc1_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 22 (all A all B -empty(unordered_pair(A,B))) # label(fc3_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 23 (all A all B subset(A,set_union2(A,B))) # label(t7_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 24 (all A (relation(A) -> (all B all C ((all D ((exists E (in(E,B) & in(ordered_pair(E,D),A))) <-> in(D,C))) <-> C = relation_image(A,B))))) # label(d13_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 25 (all A all B (ordinal(B) -> -(subset(A,B) & (all C (ordinal(C) -> -((all D (ordinal(D) -> (in(D,A) -> ordinal_subset(C,D)))) & in(C,A)))) & empty_set != A))) # label(t32_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 26 $T # label(dt_k10_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 27 (exists A (one_to_one(A) & function(A) & relation(A))) # label(rc3_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 28 (all A all B (function(B) & relation(B) -> (one_to_one(B) & in(A,relation_rng(B)) -> apply(relation_composition(function_inverse(B),B),A) = A & A = apply(B,apply(function_inverse(B),A))))) # label(t57_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 29 (all A (relation(A) -> (all B (relation(B) -> (all C (function(C) & relation(C) -> (relation_isomorphism(A,B,C) <-> relation_field(A) = relation_dom(C) & (all D all E (in(D,relation_field(A)) & in(ordered_pair(apply(C,D),apply(C,E)),B) & in(E,relation_field(A)) <-> in(ordered_pair(D,E),A))) & one_to_one(C) & relation_field(B) = relation_rng(C)))))))) # label(d7_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 30 (exists A (relation(A) & function(A))) # label(rc1_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 31 (all A all B (subset(A,B) <-> (all C (in(C,A) -> in(C,B))))) # label(d3_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 32 (all A (relation(A) -> (all B (relation_rng(A) = B <-> (all C (in(C,B) <-> (exists D in(ordered_pair(D,C),A)))))))) # label(d5_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 33 (all A all B exists C relation_of2(C,A,B)) # label(existence_m1_relset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 34 (all A all B (element(B,powerset(A)) -> element(subset_complement(A,B),powerset(A)))) # label(dt_k3_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 35 (all A (relation(A) -> (all B (relation(B) -> subset(relation_rng(relation_composition(A,B)),relation_rng(B)))))) # label(t45_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 36 (all A all B (set_difference(A,singleton(B)) = A <-> -in(B,A))) # label(t65_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 37 (all A (relation(A) -> (all B all C ((all D ((exists E (in(E,B) & in(ordered_pair(D,E),A))) <-> in(D,C))) <-> C = relation_inverse_image(A,B))))) # label(d14_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 38 (all A all B (function(B) & relation(B) -> (all C (relation(C) & function(C) -> (in(A,relation_dom(B)) -> apply(C,apply(B,A)) = apply(relation_composition(B,C),A)))))) # label(t23_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 39 (all A all B (relation(B) -> relation(relation_rng_restriction(A,B)))) # label(dt_k8_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 40 (all A (relation(A) -> (all B (relation(B) -> (subset(relation_dom(A),relation_rng(B)) -> relation_rng(relation_composition(B,A)) = relation_rng(A)))))) # label(t47_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 41 (all A all B (relation(B) -> (well_founded_relation(B) -> well_founded_relation(relation_restriction(B,A))))) # label(t31_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 42 (all A all B set_union2(A,A) = A) # label(idempotence_k2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 43 (all A all B (ordinal(B) -> (in(A,B) -> ordinal(A)))) # label(t23_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 44 (all A ((all B -(in(B,A) & (all C all D B != ordered_pair(C,D)))) <-> relation(A))) # label(d1_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 45 (all A all B -(in(A,B) & empty(B))) # label(t7_boole) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 46 (all A (relation(A) -> relation_dom(A) = relation_rng(relation_inverse(A)) & relation_rng(A) = relation_dom(relation_inverse(A)))) # label(t37_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 47 (all A all B (element(B,powerset(powerset(A))) -> -(empty_set != B & empty_set = complements_of_subsets(A,B)))) # label(t46_setfam_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 48 (all A all B ((all C (in(C,A) -> in(C,B))) -> element(A,powerset(B)))) # label(l71_subset_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 49 (all A all B (relation(B) -> (connected(B) -> connected(relation_restriction(B,A))))) # label(t23_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 50 (all A all B all C (set_intersection2(A,B) = C <-> (all D (in(D,B) & in(D,A) <-> in(D,C))))) # label(d3_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 51 (all A (relation(A) -> relation(relation_inverse(A)))) # label(dt_k4_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 52 (all A all B (element(B,powerset(powerset(A))) -> (B != empty_set -> subset_difference(A,cast_to_subset(A),union_of_subsets(A,B)) = meet_of_subsets(A,complements_of_subsets(A,B))))) # label(t47_setfam_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 53 (all A all B (relation(B) -> (antisymmetric(B) -> antisymmetric(relation_restriction(B,A))))) # label(t25_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 54 (all A (relation(A) -> (well_ordering(A) <-> well_orders(A,relation_field(A))))) # label(t8_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 55 (all A all B (relation(B) -> relation_dom_restriction(relation_rng_restriction(A,B),A) = relation_restriction(B,A))) # label(t17_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 56 (all A (relation(A) -> (all B (relation(B) -> ((all C all D (in(ordered_pair(C,D),B) <-> in(ordered_pair(C,D),A))) <-> B = A))))) # label(d2_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 57 (all A all B (disjoint(A,B) -> disjoint(B,A))) # label(symmetry_r1_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 58 $T # label(dt_k4_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 59 (all A all B all C all D -(unordered_pair(A,B) = unordered_pair(C,D) & D != A & A != C)) # label(t10_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 60 (all A all B (function(B) & relation(B) -> (all C (function(C) & relation(C) -> (in(apply(C,A),relation_dom(B)) & in(A,relation_dom(C)) <-> in(A,relation_dom(relation_composition(C,B)))))))) # label(t21_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 61 (all A set_union2(A,empty_set) = A) # label(t1_boole) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 62 (all A (empty(A) -> relation(A))) # label(cc1_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 63 (all A (relation(A) -> (connected(A) <-> is_connected_in(A,relation_field(A))))) # label(d14_relat_2) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 64 (all A all B all C (relation(C) -> relation_rng_restriction(A,relation_dom_restriction(C,B)) = relation_dom_restriction(relation_rng_restriction(A,C),B))) # label(t140_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 65 (all A (relation(A) -> (all B (is_transitive_in(A,B) <-> (all C all D all E (in(ordered_pair(D,E),A) & in(ordered_pair(C,D),A) & in(E,B) & in(D,B) & in(C,B) -> in(ordered_pair(C,E),A))))))) # label(d8_relat_2) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 66 (all A (epsilon_transitive(A) & epsilon_connected(A) <-> ordinal(A))) # label(d4_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 67 (all A all B set_union2(A,set_difference(B,A)) = set_union2(A,B)) # label(t39_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 68 (exists A (relation_empty_yielding(A) & relation(A))) # label(rc3_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 69 (all A (function(A) & relation(A) -> (one_to_one(A) -> relation_rng(function_inverse(A)) = relation_dom(A) & relation_rng(A) = relation_dom(function_inverse(A))))) # label(t55_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 70 (all A (relation(A) -> (reflexive(A) & antisymmetric(A) & well_founded_relation(A) & connected(A) & transitive(A) <-> well_ordering(A)))) # label(d4_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 71 (all A all B (function(B) & relation(B) -> (all C (function(C) & relation(C) -> (in(A,relation_dom(relation_composition(C,B))) -> apply(B,apply(C,A)) = apply(relation_composition(C,B),A)))))) # label(t22_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 72 (all A all B (element(B,powerset(powerset(A))) -> complements_of_subsets(A,complements_of_subsets(A,B)) = B)) # label(involutiveness_k7_setfam_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 73 (all A all B all C (relation(C) -> ((exists D (in(D,B) & in(ordered_pair(D,A),C) & in(D,relation_dom(C)))) <-> in(A,relation_image(C,B))))) # label(t143_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 74 (all A (relation(A) & -empty(A) -> -empty(relation_rng(A)))) # label(fc6_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 75 (all A (A = union(A) <-> being_limit_ordinal(A))) # label(d6_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 76 (all A all B (in(A,B) <-> subset(singleton(A),B))) # label(t37_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 77 (all A all B subset(set_difference(A,B),A)) # label(t36_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 78 (all A all B all C (relation(C) -> (in(A,relation_restriction(C,B)) <-> in(A,C) & in(A,cartesian_product2(B,B))))) # label(t16_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 79 (all A (empty(A) -> relation(relation_rng(A)) & empty(relation_rng(A)))) # label(fc8_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 80 (all A all B (subset(A,B) -> set_intersection2(A,B) = A)) # label(t28_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 81 $T # label(dt_k2_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 82 (all A all B all C (relation(C) -> (subset(A,B) -> subset(relation_inverse_image(C,A),relation_inverse_image(C,B))))) # label(t178_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 83 (exists A (relation(A) & empty(A) & function(A))) # label(rc2_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.31 84 (all A all B all C (relation(C) -> (in(A,relation_field(relation_restriction(C,B))) -> in(A,B) & in(A,relation_field(C))))) # label(t19_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.31 85 (all A all B all C (C = unordered_pair(A,B) <-> (all D (in(D,C) <-> B = D | A = D)))) # label(d2_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 86 (exists A -empty(A)) # label(rc2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 87 (all A all B (function(B) & relation(B) -> relation(relation_rng_restriction(A,B)) & function(relation_rng_restriction(A,B)))) # label(fc5_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 88 (all A all B (element(A,B) -> empty(B) | in(A,B))) # label(t2_subset) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 89 (all A all B subset(set_intersection2(A,B),A)) # label(t17_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 90 (all A all B set_union2(A,B) = set_union2(B,A)) # label(commutativity_k2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 91 $T # label(dt_k3_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 92 (all A (relation(A) -> (all B set_intersection2(A,cartesian_product2(B,B)) = relation_restriction(A,B)))) # label(d6_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 93 (all A all B (relation(B) -> relation_dom_restriction(B,A) = relation_composition(identity_relation(A),B))) # label(t94_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 94 (all A all B (relation(B) -> (all C (relation(C) -> (relation_rng_restriction(A,B) = C <-> (all D all E (in(ordered_pair(D,E),C) <-> in(ordered_pair(D,E),B) & in(E,A)))))))) # label(d12_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 95 (all A (relation(A) -> (empty_set = relation_rng(A) <-> relation_dom(A) = empty_set))) # label(t65_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 96 (all A (epsilon_transitive(A) -> (all B (ordinal(B) -> (proper_subset(A,B) -> in(A,B)))))) # label(t21_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 97 (all A all B (relation(B) -> (transitive(B) -> transitive(relation_restriction(B,A))))) # label(t24_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 98 (all A all B subset(A,A)) # label(reflexivity_r1_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 99 (all A (relation(A) & function(A) & empty(A) -> relation(A) & one_to_one(A) & function(A))) # label(cc2_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 100 (all A all B all C (element(C,powerset(A)) & element(B,powerset(A)) -> element(subset_difference(A,B,C),powerset(A)))) # label(dt_k6_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 101 (all A empty_set != singleton(A)) # label(l1_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 102 (all A (ordinal(A) -> (all B (ordinal(B) -> -(-in(B,A) & A != B & -in(A,B)))))) # label(t24_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 103 (all A (-empty(A) -> (exists B (-empty(B) & element(B,powerset(A)))))) # label(rc1_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 104 (all A all B (set_difference(A,B) = empty_set <-> subset(A,B))) # label(t37_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 105 (all A (relation(A) -> (all B (is_antisymmetric_in(A,B) <-> (all C all D (in(C,B) & in(D,B) & in(ordered_pair(C,D),A) & in(ordered_pair(D,C),A) -> C = D)))))) # label(d4_relat_2) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 106 (all A all B (ordinal(A) & ordinal(B) -> (subset(A,B) <-> ordinal_subset(A,B)))) # label(redefinition_r1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 107 (all A all B all C (relation_of2_as_subset(C,A,B) -> element(C,powerset(cartesian_product2(A,B))))) # label(dt_m2_relset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 108 (all A (relation(A) -> (all B (B = relation_dom(A) <-> (all C (in(C,B) <-> (exists D in(ordered_pair(C,D),A)))))))) # label(d4_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 109 (all A all B all C (relation(C) & function(C) -> (in(B,A) -> apply(C,B) = apply(relation_dom_restriction(C,A),B)))) # label(t72_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 110 (all A (relation(A) & function(A) -> (all B all C (C = relation_image(A,B) <-> (all D ((exists E (D = apply(A,E) & in(E,B) & in(E,relation_dom(A)))) <-> in(D,C))))))) # label(d12_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 111 (all A (relation(A) -> (empty_set = relation_rng(A) | relation_dom(A) = empty_set -> A = empty_set))) # label(t64_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 112 (all A all B (relation(A) & function(A) & function(B) & relation(B) -> function(relation_composition(A,B)) & relation(relation_composition(A,B)))) # label(fc1_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 113 (all A (ordinal(A) -> epsilon_connected(succ(A)) & ordinal(succ(A)) & epsilon_transitive(succ(A)) & -empty(succ(A)))) # label(fc3_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 114 (all A all B unordered_pair(A,B) = unordered_pair(B,A)) # label(commutativity_k2_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 115 (exists A (relation_empty_yielding(A) & function(A) & relation(A))) # label(rc4_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 116 (all A all B ((all C (in(C,B) <-> in(C,A))) -> B = A)) # label(t2_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 117 (all A all B -(empty(A) & empty(B) & A != B)) # label(t8_boole) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 118 (all A (relation(A) -> (is_transitive_in(A,relation_field(A)) <-> transitive(A)))) # label(d16_relat_2) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 119 (all A (relation(A) -> ((all B all C all D (in(ordered_pair(C,D),A) & in(ordered_pair(B,C),A) -> in(ordered_pair(B,D),A))) <-> transitive(A)))) # label(l2_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 120 (all A all B all C all D (in(ordered_pair(A,B),cartesian_product2(C,D)) <-> in(A,C) & in(B,D))) # label(l55_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 121 (all A all B A = set_intersection2(A,A)) # label(idempotence_k3_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 122 (all A all B (element(B,powerset(A)) -> (all C (element(C,powerset(A)) -> (disjoint(B,C) <-> subset(B,subset_complement(A,C))))))) # label(t43_subset_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 123 (all A all B (empty_set = A | A = singleton(B) <-> subset(A,singleton(B)))) # label(t39_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 124 (all A all B (element(B,powerset(A)) -> subset_complement(A,B) = set_difference(A,B))) # label(d5_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 125 (all A all B (-empty(B) & -empty(A) -> -empty(cartesian_product2(A,B)))) # label(fc4_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 126 (all A all B (-empty(A) -> -empty(set_union2(B,A)))) # label(fc3_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 127 $T # label(dt_k2_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 128 (all A set_difference(A,empty_set) = A) # label(t3_boole) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 129 (all A all B all C (subset(A,B) -> subset(set_intersection2(A,C),set_intersection2(B,C)))) # label(t26_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 130 (all A -empty(powerset(A))) # label(fc1_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 131 (all A subset(empty_set,A)) # label(t2_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 132 (all A all B (relation(A) -> relation(relation_restriction(A,B)))) # label(dt_k2_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 133 (all A all B (relation(B) -> subset(relation_rng(relation_rng_restriction(A,B)),A))) # label(t116_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 134 $T # label(dt_k3_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 135 (all A (empty(A) -> relation(relation_dom(A)) & empty(relation_dom(A)))) # label(fc7_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 136 (all A all B (element(B,powerset(powerset(A))) -> set_meet(B) = meet_of_subsets(A,B))) # label(redefinition_k6_setfam_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 137 (all A all B all C (in(A,B) & element(B,powerset(C)) -> element(A,C))) # label(t4_subset) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 138 (all A all B (-(disjoint(A,B) & (exists C (in(C,A) & in(C,B)))) & -(-disjoint(A,B) & (all C -(in(C,A) & in(C,B)))))) # label(t3_xboole_0) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 139 (all A all B (element(B,powerset(powerset(A))) -> (all C (element(C,powerset(powerset(A))) -> (complements_of_subsets(A,B) = C <-> (all D (element(D,powerset(A)) -> (in(D,C) <-> in(subset_complement(A,D),B))))))))) # label(d8_setfam_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 140 (all A (empty(A) -> ordinal(A) & epsilon_connected(A) & epsilon_transitive(A))) # label(cc3_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 141 (exists A empty(A)) # label(rc1_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 142 (all A (function(identity_relation(A)) & relation(identity_relation(A)))) # label(fc2_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 143 (all A (relation(A) -> (all B (relation(B) -> (all C (relation(C) -> (relation_composition(A,B) = C <-> (all D all E (in(ordered_pair(D,E),C) <-> (exists F (in(ordered_pair(F,E),B) & in(ordered_pair(D,F),A)))))))))))) # label(d8_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 144 (all A empty_set = set_intersection2(A,empty_set)) # label(t2_boole) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 145 (all A (relation(A) -> (all B (relation(B) -> (all C (relation(C) & function(C) -> (relation_isomorphism(A,B,C) & well_ordering(A) -> well_ordering(B)))))))) # label(t54_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 146 (all A all B (A = set_difference(A,B) <-> disjoint(A,B))) # label(t83_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 147 (all A (relation(A) -> (all B (relation(B) -> ((all C all D (in(ordered_pair(D,C),A) <-> in(ordered_pair(C,D),B))) <-> B = relation_inverse(A)))))) # label(d7_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 148 (all A exists B ((all C all D (in(C,B) & subset(D,C) -> in(D,B))) & (all C -(in(C,B) & (all D -((all E (subset(E,C) -> in(E,D))) & in(D,B))))) & (all C -(subset(C,B) & -in(C,B) & -are_equipotent(C,B))) & in(A,B))) # label(t9_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 149 (all A (relation(A) -> subset(A,cartesian_product2(relation_dom(A),relation_rng(A))))) # label(t21_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 150 (all A all B (element(B,powerset(powerset(A))) -> (empty_set != B -> subset_difference(A,cast_to_subset(A),meet_of_subsets(A,B)) = union_of_subsets(A,complements_of_subsets(A,B))))) # label(t48_setfam_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 151 (all A (function(A) & relation(A) -> (one_to_one(A) <-> (all B all C (apply(A,B) = apply(A,C) & in(C,relation_dom(A)) & in(B,relation_dom(A)) -> C = B))))) # label(d8_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 152 (all A relation(identity_relation(A))) # label(dt_k6_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 153 (all A (relation(A) -> (all B (relation(B) -> (subset(relation_rng(A),relation_dom(B)) -> relation_dom(A) = relation_dom(relation_composition(A,B))))))) # label(t46_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 154 (all A all B all C (in(A,C) & in(B,C) <-> subset(unordered_pair(A,B),C))) # label(t38_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 155 (all A (relation(A) -> (all B (relation(B) -> relation_rng(relation_composition(A,B)) = relation_image(B,relation_rng(A)))))) # label(t160_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 156 (all A (ordinal(A) -> (all B (ordinal(B) -> (in(A,B) <-> ordinal_subset(succ(A),B)))))) # label(t33_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 157 (all A all B (relation(B) -> -(A != empty_set & subset(A,relation_rng(B)) & relation_inverse_image(B,A) = empty_set))) # label(t174_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 158 (all A set_union2(A,singleton(A)) = succ(A)) # label(d1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 159 (all A all B (relation(B) -> subset(relation_rng(relation_dom_restriction(B,A)),relation_rng(B)))) # label(t99_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 160 (all A (ordinal(A) -> ((all B (ordinal(B) -> (in(B,A) -> in(succ(B),A)))) <-> being_limit_ordinal(A)))) # label(t41_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 161 (all A (function(A) & relation(A) -> (all B (relation_rng(A) = B <-> (all C (in(C,B) <-> (exists D (in(D,relation_dom(A)) & C = apply(A,D))))))))) # label(d5_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 162 (all A ((all B -in(B,A)) <-> A = empty_set)) # label(d1_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 163 (all A (epsilon_transitive(A) <-> (all B (in(B,A) -> subset(B,A))))) # label(d2_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 164 (all A all B (relation(B) -> relation_image(B,set_intersection2(relation_dom(B),A)) = relation_image(B,A))) # label(t145_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 165 (all A (relation(A) -> (all B all C (relation(C) -> (relation_dom_restriction(A,B) = C <-> (all D all E (in(ordered_pair(D,E),A) & in(D,B) <-> in(ordered_pair(D,E),C)))))))) # label(d11_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 166 (exists A (relation(A) & -empty(A))) # label(rc2_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 167 (all A all B all C (relation(C) -> (in(A,B) & in(A,relation_dom(C)) <-> in(A,relation_dom(relation_dom_restriction(C,B)))))) # label(t86_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 168 (all A (relation(A) -> (all B (relation(B) -> (all C (relation(C) & function(C) -> (relation_isomorphism(A,B,C) -> relation_isomorphism(B,A,function_inverse(C))))))))) # label(t49_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 169 (all A (ordinal(A) -> -(being_limit_ordinal(A) & (exists B (ordinal(B) & succ(B) = A))) & -(-being_limit_ordinal(A) & (all B (ordinal(B) -> succ(B) != A))))) # label(t42_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 170 (all A all B (subset(singleton(A),singleton(B)) -> B = A)) # label(t6_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 171 (all A all B (relation(B) -> (subset(A,relation_dom(B)) -> subset(A,relation_inverse_image(B,relation_image(B,A)))))) # label(t146_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 172 (all A all B (subset(A,B) <-> empty_set = set_difference(A,B))) # label(l32_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 173 (all A all B (A = empty_set | singleton(B) = A <-> subset(A,singleton(B)))) # label(l4_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 174 (all A all B all C (subset(A,B) -> subset(cartesian_product2(A,C),cartesian_product2(B,C)) & subset(cartesian_product2(C,A),cartesian_product2(C,B)))) # label(t118_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 175 (all A (function(A) & relation(A) -> function(function_inverse(A)) & relation(function_inverse(A)))) # label(dt_k2_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 176 (all A all B all C -(in(A,B) & in(C,A) & in(B,C))) # label(t3_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 177 (all A all B all C all D (in(B,D) & in(A,C) <-> in(ordered_pair(A,B),cartesian_product2(C,D)))) # label(t106_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 178 (all A (ordinal(A) -> epsilon_transitive(union(A)) & ordinal(union(A)) & epsilon_connected(union(A)))) # label(fc4_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 179 (all A (relation(A) -> (all B all C (C = fiber(A,B) <-> (all D (B != D & in(ordered_pair(D,B),A) <-> in(D,C))))))) # label(d1_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 180 (all A (relation(A) -> relation_field(A) = set_union2(relation_dom(A),relation_rng(A)))) # label(d6_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 181 (all A all B (ordinal(B) & ordinal(A) -> ordinal_subset(B,A) | ordinal_subset(A,B))) # label(connectedness_r1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 182 (all A all B all C (relation(C) & function(C) -> (apply(C,A) = B & in(A,relation_dom(C)) <-> in(ordered_pair(A,B),C)))) # label(t8_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 183 (all A all B all C (element(C,powerset(cartesian_product2(A,B))) -> relation(C))) # label(cc1_relset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 184 (all A all B (singleton(A) = B <-> (all C (in(C,B) <-> C = A)))) # label(d1_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 185 (exists A (epsilon_transitive(A) & ordinal(A) & epsilon_connected(A))) # label(rc1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 186 (all A (empty(A) -> function(A))) # label(cc1_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 187 (all A all B all C all D (relation(D) -> (in(A,C) & in(ordered_pair(A,B),D) <-> in(ordered_pair(A,B),relation_composition(identity_relation(C),D))))) # label(t74_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 188 (all A (relation(A) -> (is_reflexive_in(A,relation_field(A)) <-> reflexive(A)))) # label(d9_relat_2) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 189 (all A all B (in(A,B) <-> subset(singleton(A),B))) # label(l2_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 190 (all A all B (relation(B) -> subset(relation_dom_restriction(B,A),B))) # label(t88_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 191 (all A (relation(A) & function(A) -> (one_to_one(A) -> one_to_one(function_inverse(A))))) # label(t62_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.32 192 (all A (relation(A) -> (all B ((all C -((all D -(in(D,C) & disjoint(fiber(A,D),C))) & C != empty_set & subset(C,B))) <-> is_well_founded_in(A,B))))) # label(d3_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 193 (all A (empty(A) -> A = empty_set)) # label(t6_boole) # label(axiom) # label(non_clause). [assumption]. 1.03/1.32 194 (all A all B (relation(A) & function(A) -> function(relation_dom_restriction(A,B)) & relation(relation_dom_restriction(A,B)))) # label(fc4_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 195 (all A (function(A) & one_to_one(A) & relation(A) -> relation(relation_inverse(A)) & function(relation_inverse(A)))) # label(fc3_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 196 (all A A = union(powerset(A))) # label(t99_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 197 (all A all B all C (relation(C) & function(C) -> (in(B,relation_dom(relation_dom_restriction(C,A))) <-> in(B,A) & in(B,relation_dom(C))))) # label(l82_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 198 (all A all B (proper_subset(A,B) <-> subset(A,B) & B != A)) # label(d8_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 199 (all A all B (A = B <-> subset(B,A) & subset(A,B))) # label(d10_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 200 (all A in(A,succ(A))) # label(t10_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 201 (exists A (relation(A) & function(A) & empty(A) & epsilon_transitive(A) & ordinal(A) & epsilon_connected(A) & one_to_one(A))) # label(rc2_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 202 (all A all B (in(A,B) -> subset(A,union(B)))) # label(t92_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 203 (all A all B (-in(A,B) -> disjoint(singleton(A),B))) # label(l28_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 204 (all A (relation(A) -> (well_founded_relation(A) <-> (all B -((all C -(in(C,B) & disjoint(fiber(A,C),B))) & empty_set != B & subset(B,relation_field(A))))))) # label(d2_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 205 (all A all B (relation(B) & function(B) -> (identity_relation(A) = B <-> A = relation_dom(B) & (all C (in(C,A) -> C = apply(B,C)))))) # label(t34_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 206 $T # label(dt_k1_zfmisc_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 207 (all A (relation(A) -> relation_rng(A) = relation_image(A,relation_dom(A)))) # label(t146_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 208 (all A all B all C (unordered_pair(B,C) = singleton(A) -> C = B)) # label(t9_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 209 (all A all B exists C relation_of2_as_subset(C,A,B)) # label(existence_m2_relset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 210 $T # label(dt_k1_setfam_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 211 $T # label(dt_k2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 212 (all A all B all C (element(C,powerset(A)) & element(B,powerset(A)) -> set_difference(B,C) = subset_difference(A,B,C))) # label(redefinition_k6_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 213 (all A all B set_difference(A,B) = set_difference(set_union2(A,B),B)) # label(t40_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 214 (all A all B (element(B,powerset(powerset(A))) -> element(union_of_subsets(A,B),powerset(A)))) # label(dt_k5_setfam_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 215 (all A all B -proper_subset(A,A)) # label(irreflexivity_r2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 216 (all A all B all C (set_union2(A,B) = C <-> (all D (in(D,B) | in(D,A) <-> in(D,C))))) # label(d2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 217 $T # label(dt_k2_zfmisc_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 218 (all A (relation(A) -> ((all B all C -(in(B,relation_field(A)) & in(C,relation_field(A)) & C != B & -in(ordered_pair(B,C),A) & -in(ordered_pair(C,B),A))) <-> connected(A)))) # label(l4_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 219 (all A all B (in(A,B) -> set_union2(singleton(A),B) = B)) # label(l23_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 220 $T # label(dt_k1_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 221 $T # label(dt_k1_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 222 $T # label(dt_k1_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 223 (all A all B ((A = empty_set -> (B = empty_set <-> B = set_meet(A))) & (A != empty_set -> ((all C (in(C,B) <-> (all D (in(D,A) -> in(C,D))))) <-> set_meet(A) = B)))) # label(d1_setfam_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 224 (all A (relation(A) -> A = relation_inverse(relation_inverse(A)))) # label(involutiveness_k4_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 225 (all A all B (in(A,B) -> -in(B,A))) # label(antisymmetry_r2_hidden) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 226 (all A all B ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A))) # label(d5_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 227 (all A all B all C -(in(A,B) & empty(C) & element(B,powerset(C)))) # label(t5_subset) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 228 (all A (function(A) & relation(A) -> (one_to_one(A) -> (all B (relation(B) & function(B) -> (function_inverse(A) = B <-> (all C all D ((C = apply(A,D) & in(D,relation_dom(A)) -> in(C,relation_rng(A)) & apply(B,C) = D) & (apply(B,C) = D & in(C,relation_rng(A)) -> apply(A,D) = C & in(D,relation_dom(A))))) & relation_rng(A) = relation_dom(B))))))) # label(t54_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 229 (all A all B all C (relation(C) & function(C) -> (in(B,relation_dom(relation_dom_restriction(C,A))) -> apply(C,B) = apply(relation_dom_restriction(C,A),B)))) # label(t70_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 230 (all A all B all C (subset(A,B) & disjoint(B,C) -> disjoint(A,C))) # label(t63_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 231 (all A (function(A) & relation(A) -> (all B all C ((in(B,relation_dom(A)) -> (in(ordered_pair(B,C),A) <-> apply(A,B) = C)) & (-in(B,relation_dom(A)) -> (apply(A,B) = C <-> C = empty_set)))))) # label(d4_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 232 (all A all B set_difference(A,set_difference(A,B)) = set_intersection2(A,B)) # label(t48_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 233 (all A all B (proper_subset(A,B) -> -proper_subset(B,A))) # label(antisymmetry_r2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 234 (all A A = cast_to_subset(A)) # label(d4_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 235 (all A set_difference(empty_set,A) = empty_set) # label(t4_boole) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 236 (all A all B (subset(A,B) -> set_union2(A,B) = B)) # label(t12_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 237 (all A all B (relation(B) & relation(A) -> relation(set_difference(A,B)))) # label(fc3_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 238 (all A ((all B (in(B,A) -> ordinal(B) & subset(B,A))) -> ordinal(A))) # label(t31_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 239 (all A all B -(subset(A,B) & proper_subset(B,A))) # label(t60_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 240 (all A ((all B all C -(C != B & -in(C,B) & -in(B,C) & in(C,A) & in(B,A))) <-> epsilon_connected(A))) # label(d3_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 241 (all A all B (element(B,powerset(powerset(A))) -> element(complements_of_subsets(A,B),powerset(powerset(A))))) # label(dt_k7_setfam_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 242 (all A all B all C all D (ordered_pair(A,B) = ordered_pair(C,D) -> B = D & A = C)) # label(t33_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 243 (all A all B all C (unordered_pair(B,C) = singleton(A) -> A = B)) # label(t8_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 244 (all A exists B (in(A,B) & (all C all D (subset(D,C) & in(C,B) -> in(D,B))) & (all C -(subset(C,B) & -in(C,B) & -are_equipotent(C,B))) & (all C (in(C,B) -> in(powerset(C),B))))) # label(t136_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 245 (all A all B (-empty(A) -> -empty(set_union2(A,B)))) # label(fc2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 246 (all A all B (ordinal(A) & ordinal(B) -> ordinal_subset(A,A))) # label(reflexivity_r1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 247 (all A all B all C (element(C,powerset(A)) -> -(in(B,C) & in(B,subset_complement(A,C))))) # label(t54_subset_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 248 (all A -empty(singleton(A))) # label(fc2_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 249 (all A (relation(A) -> (all B (relation(B) -> ((all C all D (in(ordered_pair(C,D),A) -> in(ordered_pair(C,D),B))) <-> subset(A,B)))))) # label(d3_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 250 (all A all B (element(B,powerset(powerset(A))) -> element(meet_of_subsets(A,B),powerset(A)))) # label(dt_k6_setfam_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 251 (all A all B (relation(A) & relation(B) -> relation(set_intersection2(A,B)))) # label(fc1_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 252 (all A (relation(A) -> ((all B all C (in(ordered_pair(C,B),A) & in(ordered_pair(B,C),A) -> C = B)) <-> antisymmetric(A)))) # label(l3_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 253 (all A all B (relation(B) -> subset(relation_inverse_image(B,A),relation_dom(B)))) # label(t167_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 254 (all A all B all C (subset(A,B) -> subset(set_difference(A,C),set_difference(B,C)))) # label(t33_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 255 (all A all B -((all C -((all D -(in(D,B) & in(D,C))) & in(C,B))) & in(A,B))) # label(t7_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 256 (all A all B (relation(B) & function(B) -> (all C (relation(C) & function(C) -> (set_intersection2(relation_dom(C),A) = relation_dom(B) & (all D (in(D,relation_dom(B)) -> apply(C,D) = apply(B,D))) <-> B = relation_dom_restriction(C,A)))))) # label(t68_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 257 (all A (relation(A) -> ((all B all C -in(ordered_pair(B,C),A)) -> A = empty_set))) # label(t56_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 258 (all A all B (relation(B) & relation(A) -> relation(relation_composition(A,B)))) # label(dt_k5_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 259 (all A (relation(A) -> (all B (relation(B) -> subset(relation_dom(relation_composition(A,B)),relation_dom(A)))))) # label(t44_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 260 (all A all B (relation(B) -> subset(relation_field(relation_restriction(B,A)),A) & subset(relation_field(relation_restriction(B,A)),relation_field(B)))) # label(t20_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 261 (all A all B -empty(ordered_pair(A,B))) # label(fc1_zfmisc_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 262 (all A all B all C all D ((all E (in(E,D) <-> -(E != C & E != B & A != E))) <-> unordered_triple(A,B,C) = D)) # label(d1_enumset1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 263 (all A all B all C (relation_of2_as_subset(C,A,B) <-> relation_of2(C,A,B))) # label(redefinition_m2_relset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 264 (all A all B (in(A,B) -> subset(A,union(B)))) # label(l50_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 265 (all A (relation(A) -> (antisymmetric(A) <-> is_antisymmetric_in(A,relation_field(A))))) # label(d12_relat_2) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 266 (all A all B (relation(B) -> relation_restriction(B,A) = relation_rng_restriction(A,relation_dom_restriction(B,A)))) # label(t18_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 267 (all A all B all C (subset(C,B) & subset(A,B) -> subset(set_union2(A,C),B))) # label(t8_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 268 (all A (empty(A) -> empty(relation_inverse(A)) & relation(relation_inverse(A)))) # label(fc11_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 269 (all A all B ((empty(A) -> (element(B,A) <-> empty(B))) & (-empty(A) -> (element(B,A) <-> in(B,A))))) # label(d2_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 270 $T # label(dt_k1_enumset1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 271 (all A all B all C ((all D ((exists E exists F (in(E,A) & in(F,B) & ordered_pair(E,F) = D)) <-> in(D,C))) <-> C = cartesian_product2(A,B))) # label(d2_zfmisc_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 272 (all A all B (relation(B) & empty(A) -> empty(relation_composition(A,B)) & relation(relation_composition(A,B)))) # label(fc9_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 273 (all A (function(A) & relation(A) -> (all B all C ((all D (in(D,C) <-> in(D,relation_dom(A)) & in(apply(A,D),B))) <-> C = relation_inverse_image(A,B))))) # label(d13_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 274 (all A (relation(A) -> (all B (relation(B) -> (all C (function(C) & relation(C) -> (relation_isomorphism(A,B,C) -> (connected(A) -> connected(B)) & (well_founded_relation(A) -> well_founded_relation(B)) & (antisymmetric(A) -> antisymmetric(B)) & (transitive(A) -> transitive(B)) & (reflexive(A) -> reflexive(B))))))))) # label(t53_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 275 $T # label(dt_k1_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 276 $T # label(dt_m1_relset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 277 (all A all B (in(A,B) -> element(A,B))) # label(t1_subset) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 278 (all A all B (relation(A) -> relation(relation_dom_restriction(A,B)))) # label(dt_k7_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 279 (all A exists B element(B,A)) # label(existence_m1_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 280 (all A element(cast_to_subset(A),powerset(A))) # label(dt_k2_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 281 (all A all B (relation(B) -> subset(relation_rng_restriction(A,B),B))) # label(t117_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 282 (all A (relation(A) -> (all B ((all C all D -(in(C,B) & in(D,B) & -in(ordered_pair(D,C),A) & -in(ordered_pair(C,D),A) & D != C)) <-> is_connected_in(A,B))))) # label(d6_relat_2) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 283 (all A all B all C (subset(A,B) & subset(A,C) -> subset(A,set_intersection2(B,C)))) # label(t19_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 284 (all A all B (in(B,A) -> apply(identity_relation(A),B) = B)) # label(t35_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 285 (all A all B (relation(B) -> subset(relation_image(B,A),relation_rng(B)))) # label(t144_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 286 (all A all B set_intersection2(A,B) = set_intersection2(B,A)) # label(commutativity_k3_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 287 (all A all B (relation(B) -> set_intersection2(relation_dom(B),A) = relation_dom(relation_dom_restriction(B,A)))) # label(t90_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 288 (all A exists B (empty(B) & element(B,powerset(A)))) # label(rc2_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 289 $T # label(dt_k1_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 290 (all A -empty(succ(A))) # label(fc1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 291 (all A all B (relation(A) & relation(B) -> relation(set_union2(A,B)))) # label(fc2_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 292 (all A (relation(A) -> (all B (is_reflexive_in(A,B) & is_connected_in(A,B) & is_well_founded_in(A,B) & is_antisymmetric_in(A,B) & is_transitive_in(A,B) <-> well_orders(A,B))))) # label(d5_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 293 $T # label(dt_k9_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 294 (all A all B (relation(B) & function(B) -> (subset(A,relation_rng(B)) -> A = relation_image(B,relation_inverse_image(B,A))))) # label(t147_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 295 (all A all B (-((exists C in(C,set_intersection2(A,B))) & disjoint(A,B)) & -((all C -in(C,set_intersection2(A,B))) & -disjoint(A,B)))) # label(t4_xboole_0) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 296 (all A (epsilon_transitive(A) & epsilon_connected(A) -> ordinal(A))) # label(cc2_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 297 (all A all B (relation_empty_yielding(A) & relation(A) -> relation(relation_dom_restriction(A,B)) & relation_empty_yielding(relation_dom_restriction(A,B)))) # label(fc13_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 298 (all A all B (relation(B) -> relation_rng(relation_rng_restriction(A,B)) = set_intersection2(relation_rng(B),A))) # label(t119_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 299 (all A all B (subset(A,B) -> set_union2(A,set_difference(B,A)) = B)) # label(t45_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 300 (all A all B all C (relation(C) -> (in(ordered_pair(A,B),C) -> in(B,relation_rng(C)) & in(A,relation_dom(C))))) # label(t20_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 301 $T # label(dt_k3_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 302 (all A all B (function(B) & relation(B) -> subset(relation_image(B,relation_inverse_image(B,A)),A))) # label(t145_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 303 (all A all B all C (relation(C) -> (in(ordered_pair(A,B),C) -> in(B,relation_field(C)) & in(A,relation_field(C))))) # label(t30_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 304 (exists A (-empty(A) & ordinal(A) & epsilon_connected(A) & epsilon_transitive(A))) # label(rc3_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 305 (all A all B all C (relation(C) -> subset(fiber(relation_restriction(C,A),B),fiber(C,B)))) # label(t21_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 306 (all A all B (relation(B) -> (well_ordering(B) -> well_ordering(relation_restriction(B,A))))) # label(t32_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.33 307 (all A all B (powerset(A) = B <-> (all C (subset(C,A) <-> in(C,B))))) # label(d1_zfmisc_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.33 308 (all A all B (relation(B) -> (reflexive(B) -> reflexive(relation_restriction(B,A))))) # label(t22_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.39 309 (all A all B all C all D (subset(C,D) & subset(A,B) -> subset(cartesian_product2(A,C),cartesian_product2(B,D)))) # label(t119_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.39 310 (all A (relation(A) & -empty(A) -> -empty(relation_dom(A)))) # label(fc5_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.39 311 (all A all B (relation(B) -> (well_ordering(B) & subset(A,relation_field(B)) -> A = relation_field(relation_restriction(B,A))))) # label(t39_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.39 312 (all A all B (element(A,powerset(B)) <-> subset(A,B))) # label(t3_subset) # label(axiom) # label(non_clause). [assumption]. 1.03/1.39 313 (all A all B (B = union(A) <-> (all C (in(C,B) <-> (exists D (in(C,D) & in(D,A))))))) # label(d4_tarski) # label(axiom) # label(non_clause). [assumption]. 1.03/1.39 314 (all A all B (element(B,powerset(powerset(A))) -> union_of_subsets(A,B) = union(B))) # label(redefinition_k5_setfam_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.39 315 (all A all B (disjoint(A,B) <-> empty_set = set_intersection2(A,B))) # label(d7_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.03/1.39 316 (all A unordered_pair(A,A) = singleton(A)) # label(t69_enumset1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.39 317 (all A (ordinal(A) -> epsilon_transitive(A) & epsilon_connected(A))) # label(cc1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.39 318 (all A (relation(A) & function(A) -> (one_to_one(A) -> relation_inverse(A) = function_inverse(A)))) # label(d9_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.39 319 $T # label(dt_m1_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.03/1.39 320 (all A all B -(disjoint(singleton(A),B) & in(A,B))) # label(l25_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.39 321 (all A (relation(A) -> (is_well_founded_in(A,relation_field(A)) <-> well_founded_relation(A)))) # label(t5_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.03/1.39 322 -(all A all B all C (relation_of2_as_subset(C,A,B) -> subset(relation_dom(C),A) & subset(relation_rng(C),B))) # label(t12_relset_1) # label(negated_conjecture) # label(non_clause). [assumption]. 1.03/1.39 1.03/1.39 ============================== end of process non-clausal formulas === 1.03/1.39 1.03/1.39 ============================== PROCESS INITIAL CLAUSES =============== 1.03/1.39 1.03/1.39 ============================== PREDICATE ELIMINATION ================= 1.03/1.39 323 relation_of2_as_subset(A,B,C) | -relation_of2(A,B,C) # label(redefinition_m2_relset_1) # label(axiom). [clausify(263)]. 1.03/1.39 324 relation_of2(f17(A,B),A,B) # label(existence_m1_relset_1) # label(axiom). [clausify(33)]. 1.03/1.39 325 -relation_of2_as_subset(A,B,C) | relation_of2(A,B,C) # label(redefinition_m2_relset_1) # label(axiom). [clausify(263)]. 1.03/1.39 Derived: relation_of2_as_subset(f17(A,B),A,B). [resolve(323,b,324,a)]. 1.03/1.39 326 epsilon_transitive(A) | -ordinal(A) # label(d4_ordinal1) # label(axiom). [clausify(66)]. 1.03/1.39 327 -epsilon_transitive(A) | -epsilon_connected(A) | ordinal(A) # label(d4_ordinal1) # label(axiom). [clausify(66)]. 1.03/1.39 328 -epsilon_transitive(A) | -ordinal(B) | -proper_subset(A,B) | in(A,B) # label(t21_ordinal1) # label(lemma). [clausify(96)]. 1.03/1.39 Derived: -ordinal(A) | -proper_subset(B,A) | in(B,A) | -ordinal(B). [resolve(328,a,326,a)]. 1.03/1.39 329 -ordinal(A) | epsilon_transitive(succ(A)) # label(fc3_ordinal1) # label(axiom). [clausify(113)]. 1.03/1.39 Derived: -ordinal(A) | -epsilon_connected(succ(A)) | ordinal(succ(A)). [resolve(329,b,327,a)]. 1.03/1.39 Derived: -ordinal(A) | -ordinal(B) | -proper_subset(succ(A),B) | in(succ(A),B). [resolve(329,b,328,a)]. 1.03/1.39 330 -empty(A) | epsilon_transitive(A) # label(cc3_ordinal1) # label(axiom). [clausify(140)]. 1.03/1.39 Derived: -empty(A) | -epsilon_connected(A) | ordinal(A). [resolve(330,b,327,a)]. 1.03/1.39 Derived: -empty(A) | -ordinal(B) | -proper_subset(A,B) | in(A,B). [resolve(330,b,328,a)]. 1.03/1.39 331 -epsilon_transitive(A) | -in(B,A) | subset(B,A) # label(d2_ordinal1) # label(axiom). [clausify(163)]. 1.03/1.39 Derived: -in(A,B) | subset(A,B) | -ordinal(B). [resolve(331,a,326,a)]. 1.03/1.42 Derived: -in(A,succ(B)) | subset(A,succ(B)) | -ordinal(B). [resolve(331,a,329,b)]. 1.03/1.42 Derived: -in(A,B) | subset(A,B) | -empty(B). [resolve(331,a,330,b)]. 1.03/1.42 332 epsilon_transitive(A) | in(f65(A),A) # label(d2_ordinal1) # label(axiom). [clausify(163)]. 1.03/1.42 Derived: in(f65(A),A) | -epsilon_connected(A) | ordinal(A). [resolve(332,a,327,a)]. 1.03/1.42 Derived: in(f65(A),A) | -ordinal(B) | -proper_subset(A,B) | in(A,B). [resolve(332,a,328,a)]. 1.03/1.42 Derived: in(f65(A),A) | -in(B,A) | subset(B,A). [resolve(332,a,331,a)]. 1.03/1.42 333 epsilon_transitive(A) | -subset(f65(A),A) # label(d2_ordinal1) # label(axiom). [clausify(163)]. 1.03/1.42 Derived: -subset(f65(A),A) | -epsilon_connected(A) | ordinal(A). [resolve(333,a,327,a)]. 1.03/1.42 Derived: -subset(f65(A),A) | -ordinal(B) | -proper_subset(A,B) | in(A,B). [resolve(333,a,328,a)]. 1.03/1.42 Derived: -subset(f65(A),A) | -in(B,A) | subset(B,A). [resolve(333,a,331,a)]. 1.03/1.42 334 -ordinal(A) | epsilon_transitive(union(A)) # label(fc4_ordinal1) # label(axiom). [clausify(178)]. 1.03/1.42 Derived: -ordinal(A) | -ordinal(B) | -proper_subset(union(A),B) | in(union(A),B). [resolve(334,b,328,a)]. 1.03/1.42 Derived: -ordinal(A) | -in(B,union(A)) | subset(B,union(A)). [resolve(334,b,331,a)]. 1.03/1.42 335 epsilon_transitive(empty_set) # label(fc2_ordinal1_AndRHS_AndRHS_AndRHS_AndRHS_AndLHS) # label(axiom). [assumption]. 1.03/1.42 Derived: -epsilon_connected(empty_set) | ordinal(empty_set). [resolve(335,a,327,a)]. 1.03/1.42 Derived: -ordinal(A) | -proper_subset(empty_set,A) | in(empty_set,A). [resolve(335,a,328,a)]. 1.03/1.42 Derived: -in(A,empty_set) | subset(A,empty_set). [resolve(335,a,331,a)]. 1.03/1.42 336 epsilon_transitive(c10) # label(rc1_ordinal1) # label(axiom). [clausify(185)]. 1.03/1.42 Derived: -ordinal(A) | -proper_subset(c10,A) | in(c10,A). [resolve(336,a,328,a)]. 1.03/1.42 Derived: -in(A,c10) | subset(A,c10). [resolve(336,a,331,a)]. 1.03/1.42 337 epsilon_transitive(c11) # label(rc2_ordinal1) # label(axiom). [clausify(201)]. 1.03/1.42 Derived: -ordinal(A) | -proper_subset(c11,A) | in(c11,A). [resolve(337,a,328,a)]. 1.03/1.42 Derived: -in(A,c11) | subset(A,c11). [resolve(337,a,331,a)]. 1.03/1.42 338 -epsilon_transitive(A) | -epsilon_connected(A) | ordinal(A) # label(cc2_ordinal1) # label(axiom). [clausify(296)]. 1.03/1.42 Derived: -epsilon_connected(union(A)) | ordinal(union(A)) | -ordinal(A). [resolve(338,a,334,b)]. 1.03/1.42 Derived: -epsilon_connected(c10) | ordinal(c10). [resolve(338,a,336,a)]. 1.03/1.42 Derived: -epsilon_connected(c11) | ordinal(c11). [resolve(338,a,337,a)]. 1.03/1.42 339 epsilon_transitive(c12) # label(rc3_ordinal1) # label(axiom). [clausify(304)]. 1.03/1.42 Derived: -epsilon_connected(c12) | ordinal(c12). [resolve(339,a,327,a)]. 1.03/1.42 Derived: -ordinal(A) | -proper_subset(c12,A) | in(c12,A). [resolve(339,a,328,a)]. 1.03/1.42 Derived: -in(A,c12) | subset(A,c12). [resolve(339,a,331,a)]. 1.03/1.42 340 -ordinal(A) | epsilon_transitive(A) # label(cc1_ordinal1) # label(axiom). [clausify(317)]. 1.03/1.42 341 union(A) = A | -being_limit_ordinal(A) # label(d6_ordinal1) # label(axiom). [clausify(75)]. 1.03/1.42 342 union(A) != A | being_limit_ordinal(A) # label(d6_ordinal1) # label(axiom). [clausify(75)]. 1.03/1.42 343 -ordinal(A) | ordinal(f60(A)) | being_limit_ordinal(A) # label(t41_ordinal1) # label(lemma). [clausify(160)]. 1.03/1.42 Derived: -ordinal(A) | ordinal(f60(A)) | union(A) = A. [resolve(343,c,341,b)]. 1.03/1.42 344 -ordinal(A) | in(f60(A),A) | being_limit_ordinal(A) # label(t41_ordinal1) # label(lemma). [clausify(160)]. 1.03/1.42 Derived: -ordinal(A) | in(f60(A),A) | union(A) = A. [resolve(344,c,341,b)]. 1.03/1.42 345 -ordinal(A) | -in(succ(f60(A)),A) | being_limit_ordinal(A) # label(t41_ordinal1) # label(lemma). [clausify(160)]. 1.03/1.42 Derived: -ordinal(A) | -in(succ(f60(A)),A) | union(A) = A. [resolve(345,c,341,b)]. 1.03/1.42 346 -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A) | -being_limit_ordinal(A) # label(t41_ordinal1) # label(lemma). [clausify(160)]. 1.03/1.42 Derived: -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A) | union(A) != A. [resolve(346,e,342,b)]. 1.03/1.42 Derived: -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A) | -ordinal(A) | ordinal(f60(A)). [resolve(346,e,343,c)]. 1.03/1.42 Derived: -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A) | -ordinal(A) | in(f60(A),A). [resolve(346,e,344,c)]. 1.03/1.42 Derived: -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A) | -ordinal(A) | -in(succ(f60(A)),A). [resolve(346,e,345,c)]. 1.47/1.74 347 -ordinal(A) | -being_limit_ordinal(A) | -ordinal(B) | succ(B) != A # label(t42_ordinal1) # label(lemma). [clausify(169)]. 1.47/1.74 Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | union(A) != A. [resolve(347,b,342,b)]. 1.47/1.74 Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | -ordinal(A) | ordinal(f60(A)). [resolve(347,b,343,c)]. 1.47/1.74 Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | -ordinal(A) | in(f60(A),A). [resolve(347,b,344,c)]. 1.47/1.74 Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | -ordinal(A) | -in(succ(f60(A)),A). [resolve(347,b,345,c)]. 1.47/1.74 348 -ordinal(A) | being_limit_ordinal(A) | ordinal(f68(A)) # label(t42_ordinal1) # label(lemma). [clausify(169)]. 1.47/1.74 Derived: -ordinal(A) | ordinal(f68(A)) | union(A) = A. [resolve(348,b,341,b)]. 1.47/1.74 Derived: -ordinal(A) | ordinal(f68(A)) | -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A). [resolve(348,b,346,e)]. 1.47/1.74 Derived: -ordinal(A) | ordinal(f68(A)) | -ordinal(A) | -ordinal(B) | succ(B) != A. [resolve(348,b,347,b)]. 1.47/1.74 349 -ordinal(A) | being_limit_ordinal(A) | succ(f68(A)) = A # label(t42_ordinal1) # label(lemma). [clausify(169)]. 1.47/1.74 Derived: -ordinal(A) | succ(f68(A)) = A | union(A) = A. [resolve(349,b,341,b)]. 1.47/1.74 Derived: -ordinal(A) | succ(f68(A)) = A | -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A). [resolve(349,b,346,e)]. 1.47/1.74 Derived: -ordinal(A) | succ(f68(A)) = A | -ordinal(A) | -ordinal(B) | succ(B) != A. [resolve(349,b,347,b)]. 1.47/1.74 350 relation_of2_as_subset(f76(A,B),A,B) # label(existence_m2_relset_1) # label(axiom). [clausify(209)]. 1.47/1.74 351 -relation_of2_as_subset(A,B,C) | element(A,powerset(cartesian_product2(B,C))) # label(dt_m2_relset_1) # label(axiom). [clausify(107)]. 1.47/1.74 Derived: element(f76(A,B),powerset(cartesian_product2(A,B))). [resolve(350,a,351,a)]. 1.47/1.74 352 relation_of2_as_subset(c15,c13,c14) # label(t12_relset_1) # label(negated_conjecture). [clausify(322)]. 1.47/1.74 Derived: element(c15,powerset(cartesian_product2(c13,c14))). [resolve(352,a,351,a)]. 1.47/1.74 353 relation_of2_as_subset(f17(A,B),A,B). [resolve(323,b,324,a)]. 1.47/1.74 Derived: element(f17(A,B),powerset(cartesian_product2(A,B))). [resolve(353,a,351,a)]. 1.47/1.74 1.47/1.74 ============================== end predicate elimination ============= 1.47/1.74 1.47/1.74 Auto_denials: (non-Horn, no changes). 1.47/1.74 1.47/1.74 Term ordering decisions: 1.47/1.74 Function symbol KB weights: empty_set=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. c9=1. c10=1. c11=1. c12=1. c13=1. c14=1. c15=1. ordered_pair=1. apply=1. relation_dom_restriction=1. cartesian_product2=1. relation_composition=1. set_difference=1. set_intersection2=1. relation_image=1. set_union2=1. relation_inverse_image=1. relation_rng_restriction=1. relation_restriction=1. unordered_pair=1. complements_of_subsets=1. fiber=1. subset_complement=1. meet_of_subsets=1. union_of_subsets=1. f3=1. f4=1. f5=1. f10=1. f13=1. f15=1. f16=1. f17=1. f22=1. f23=1. f24=1. f26=1. f27=1. f28=1. f29=1. f30=1. f36=1. f37=1. f39=1. f40=1. f44=1. f48=1. f54=1. f55=1. f57=1. f62=1. f63=1. f70=1. f71=1. f73=1. f75=1. f76=1. f80=1. f81=1. f83=1. f84=1. f89=1. f90=1. f93=1. f105=1. f106=1. f108=1. f109=1. f111=1. f112=1. relation_dom=1. powerset=1. relation_rng=1. relation_field=1. singleton=1. succ=1. union=1. identity_relation=1. function_inverse=1. relation_inverse=1. set_meet=1. cast_to_subset=1. f2=1. f21=1. f35=1. f45=1. f46=1. f47=1. f56=1. f58=1. f59=1. f60=1. f64=1. f65=1. f68=1. f74=1. f78=1. f79=1. f85=1. f86=1. f87=1. f88=1. f91=1. f92=1. f95=1. f96=1. f104=1. f107=1. unordered_triple=1. subset_difference=1. f1=1. f6=1. f7=1. f8=1. f11=1. f12=1. f14=1. f18=1. f19=1. f25=1. f31=1. f32=1. f33=1. f34=1. f38=1. f42=1. f43=1. f49=1. f51=1. f52=1. f53=1. f61=1. f66=1. f67=1. f69=1. f72=1. f77=1. f82=1. f94=1. f98=1. f99=1. f100=1. f103=1. f110=1. f9=1. f20=1. f41=1. f97=1. f101=1. f102=1. f50=1. 1.47/1.74 1.47/1.74 ============================== end of process initial clauses ======== 1.47/1.74 1.47/1.74 ============================== CLAUSES FOR SEARCH ==================== 1.47/1.74 1.47/1.74 ============================== end of clauses for search ============= 1.47/1.74 1.47/1.74 ============================== SEARCH ========================Alarm clock 119.79/120.10 Prover9 interrupted 119.79/120.10 EOF