0.07/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.13 % Command : tptp2X_and_run_prover9 %d %s 0.13/0.34 % Computer : n011.cluster.edu 0.13/0.34 % Model : x86_64 x86_64 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.34 % Memory : 8042.1875MB 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.34 % CPULimit : 960 0.13/0.34 % WCLimit : 120 0.13/0.34 % DateTime : Tue Aug 9 06:28:06 EDT 2022 0.13/0.34 % CPUTime : 0.96/1.29 ============================== Prover9 =============================== 0.96/1.29 Prover9 (32) version 2009-11A, November 2009. 0.96/1.29 Process 31351 was started by sandbox on n011.cluster.edu, 0.96/1.29 Tue Aug 9 06:28:07 2022 0.96/1.29 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 960 -f /tmp/Prover9_31198_n011.cluster.edu". 0.96/1.29 ============================== end of head =========================== 0.96/1.29 0.96/1.29 ============================== INPUT ================================= 0.96/1.29 0.96/1.29 % Reading from file /tmp/Prover9_31198_n011.cluster.edu 0.96/1.29 0.96/1.29 set(prolog_style_variables). 0.96/1.29 set(auto2). 0.96/1.29 % set(auto2) -> set(auto). 0.96/1.29 % set(auto) -> set(auto_inference). 0.96/1.29 % set(auto) -> set(auto_setup). 0.96/1.29 % set(auto_setup) -> set(predicate_elim). 0.96/1.29 % set(auto_setup) -> assign(eq_defs, unfold). 0.96/1.29 % set(auto) -> set(auto_limits). 0.96/1.29 % set(auto_limits) -> assign(max_weight, "100.000"). 0.96/1.29 % set(auto_limits) -> assign(sos_limit, 20000). 0.96/1.29 % set(auto) -> set(auto_denials). 0.96/1.29 % set(auto) -> set(auto_process). 0.96/1.29 % set(auto2) -> assign(new_constants, 1). 0.96/1.29 % set(auto2) -> assign(fold_denial_max, 3). 0.96/1.29 % set(auto2) -> assign(max_weight, "200.000"). 0.96/1.29 % set(auto2) -> assign(max_hours, 1). 0.96/1.29 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.96/1.29 % set(auto2) -> assign(max_seconds, 0). 0.96/1.29 % set(auto2) -> assign(max_minutes, 5). 0.96/1.29 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.96/1.29 % set(auto2) -> set(sort_initial_sos). 0.96/1.29 % set(auto2) -> assign(sos_limit, -1). 0.96/1.29 % set(auto2) -> assign(lrs_ticks, 3000). 0.96/1.29 % set(auto2) -> assign(max_megs, 400). 0.96/1.29 % set(auto2) -> assign(stats, some). 0.96/1.29 % set(auto2) -> clear(echo_input). 0.96/1.29 % set(auto2) -> set(quiet). 0.96/1.29 % set(auto2) -> clear(print_initial_clauses). 0.96/1.29 % set(auto2) -> clear(print_given). 0.96/1.29 assign(lrs_ticks,-1). 0.96/1.29 assign(sos_limit,10000). 0.96/1.29 assign(order,kbo). 0.96/1.29 set(lex_order_vars). 0.96/1.29 clear(print_given). 0.96/1.29 0.96/1.29 % formulas(sos). % not echoed (382 formulas) 0.96/1.29 0.96/1.29 ============================== end of input ========================== 0.96/1.29 0.96/1.29 % From the command line: assign(max_seconds, 960). 0.96/1.29 0.96/1.29 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.96/1.29 0.96/1.29 % Formulas that are not ordinary clauses: 0.96/1.29 1 $T # label(dt_k1_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 2 (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(lemma) # label(non_clause). [assumption]. 0.96/1.29 3 (all A exists B element(B,A)) # label(existence_m1_subset_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 4 (all A all B set_union2(A,B) = set_union2(A,set_difference(B,A))) # label(t39_xboole_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 5 (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]. 0.96/1.29 6 (all A all B (function(A) & relation(A) -> function(relation_dom_restriction(A,B)) & relation(relation_dom_restriction(A,B)))) # label(fc4_funct_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 7 (all A all B (disjoint(A,B) -> disjoint(B,A))) # label(symmetry_r1_xboole_0) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 8 (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]. 0.96/1.29 9 $T # label(dt_k1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 10 (all A all B (empty_set = set_difference(A,B) <-> subset(A,B))) # label(t37_xboole_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 11 (all A all B subset(A,set_union2(A,B))) # label(t7_xboole_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 12 (all A all B all C all D (relation_of2_as_subset(D,C,A) -> (subset(relation_rng(D),B) -> relation_of2_as_subset(D,C,B)))) # label(t14_relset_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 13 $T # label(dt_k9_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 14 (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),B) & in(E,A) <-> in(ordered_pair(D,E),C)))))))) # label(d12_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 15 (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]. 0.96/1.29 16 (all A all B (ordinal(B) -> ((all C all D all E (C = D & E = C & (exists G (ordinal(G) & in(G,A) & G = E)) & (exists F (ordinal(F) & F = D & in(F,A))) -> D = E)) -> (exists C all D (in(D,C) <-> (exists E (in(E,succ(B)) & E = D & (exists H (H = D & in(H,A) & ordinal(H)))))))))) # label(s1_tarski__e8_6__wellord2__1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 17 (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]. 0.96/1.29 18 (all A (relation(A) & function(A) -> (all B ((all C ((exists D (in(D,relation_dom(A)) & C = apply(A,D))) <-> in(C,B))) <-> relation_rng(A) = B)))) # label(d5_funct_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 19 (all A all B subset(set_intersection2(A,B),A)) # label(t17_xboole_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 20 (all A all B (relation(B) -> -(well_ordering(B) & (all C (relation(C) -> -well_orders(C,A))) & equipotent(A,relation_field(B))))) # label(l30_wellord2) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 21 (all A all B all C (cartesian_product2(A,B) = C <-> (all D (in(D,C) <-> (exists E exists F (in(E,A) & in(F,B) & D = ordered_pair(E,F))))))) # label(d2_zfmisc_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 22 (all A singleton(A) != empty_set) # label(l1_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 23 (all A (relation(A) -> relation_rng(A) = relation_dom(relation_inverse(A)) & relation_dom(A) = relation_rng(relation_inverse(A)))) # label(t37_relat_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 24 (all A all B -empty(unordered_pair(A,B))) # label(fc3_subset_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 25 (all A (relation(A) <-> (all B -((all C all D B != ordered_pair(C,D)) & in(B,A))))) # label(d1_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 26 (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]. 0.96/1.29 27 (all A (ordinal(A) -> (being_limit_ordinal(A) <-> (all B (ordinal(B) -> (in(B,A) -> in(succ(B),A))))))) # label(t41_ordinal1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 28 (all A all B (subset(A,B) -> B = set_union2(A,set_difference(B,A)))) # label(t45_xboole_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 29 (all A all B all C (relation_of2_as_subset(C,B,A) -> (B = relation_dom_as_subset(B,A,C) <-> (all D -((all E -in(ordered_pair(D,E),C)) & in(D,B)))))) # label(t22_relset_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 30 (all A all B (subset(A,singleton(B)) <-> A = empty_set | A = singleton(B))) # label(l4_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 31 (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]. 0.96/1.29 32 (all A (empty(A) -> empty(relation_inverse(A)) & relation(relation_inverse(A)))) # label(fc11_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 33 (all A (ordinal(A) -> connected(inclusion_relation(A)))) # label(t4_wellord2) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 34 (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]. 0.96/1.29 35 (all A all B (element(B,powerset(A)) -> (all C (element(C,powerset(A)) -> (subset(B,subset_complement(A,C)) <-> disjoint(B,C)))))) # label(t43_subset_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 36 (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]. 0.96/1.29 37 (all A exists B all C (in(C,B) <-> in(C,A) & ordinal(C))) # label(s1_xboole_0__e6_22__wellord2) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 38 (all A all B all C (subset(unordered_pair(A,B),C) <-> in(B,C) & in(A,C))) # label(t38_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 39 (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]. 0.96/1.29 40 (all A all B all C all D -(unordered_pair(C,D) = unordered_pair(A,B) & A != D & A != C)) # label(t10_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 41 (all A all B all C ((all D (in(D,C) <-> in(D,B) | in(D,A))) <-> C = set_union2(A,B))) # label(d2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 42 (all A unordered_pair(A,A) = singleton(A)) # label(t69_enumset1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 43 (all A all B (subset(A,B) <-> set_difference(A,B) = empty_set)) # label(l32_xboole_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 44 (all A (relation(A) -> (all B all C ((all D (in(ordered_pair(D,B),A) & D != B <-> in(D,C))) <-> C = fiber(A,B))))) # label(d1_wellord1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 45 (all A all B (relation(B) -> subset(relation_rng_restriction(A,B),B))) # label(t117_relat_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 46 $T # label(dt_m1_relset_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 47 (all A (relation(A) -> ((all B -(subset(B,relation_field(A)) & (all C -(in(C,B) & disjoint(fiber(A,C),B))) & B != empty_set)) <-> well_founded_relation(A)))) # label(d2_wellord1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 48 (all A all B (relation(B) & function(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]. 0.96/1.29 49 (all A (relation(A) & function(A) -> (all B all C (C = relation_inverse_image(A,B) <-> (all D (in(apply(A,D),B) & in(D,relation_dom(A)) <-> in(D,C))))))) # label(d13_funct_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 50 $T # label(dt_k1_tarski) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 51 $T # label(dt_k4_xboole_0) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 52 (all A all B all C (relation_of2(C,A,B) -> relation_dom_as_subset(A,B,C) = relation_dom(C))) # label(redefinition_k4_relset_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 53 (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]. 0.96/1.29 54 (all A (relation(A) -> relation_rng(A) = relation_image(A,relation_dom(A)))) # label(t146_relat_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 55 $T # label(dt_k1_funct_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 56 (all A (relation(A) -> (all B (relation(B) -> (all C (function(C) & relation(C) -> (relation_isomorphism(A,B,C) -> relation_isomorphism(B,A,function_inverse(C))))))))) # label(t49_wellord1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 57 (all A all B all C all D ((all E (in(E,D) <-> -(B != E & C != E & E != A))) <-> unordered_triple(A,B,C) = D)) # label(d1_enumset1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 58 (all A all B subset(A,A)) # label(reflexivity_r1_tarski) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 59 (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]. 0.96/1.29 60 (all A all B exists C relation_of2_as_subset(C,A,B)) # label(existence_m2_relset_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 61 (all A (empty(A) -> A = empty_set)) # label(t6_boole) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 62 (exists A (relation(A) & function(A))) # label(rc1_funct_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 63 (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]. 0.96/1.29 64 (exists A (function(A) & one_to_one(A) & relation(A))) # label(rc3_funct_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 65 (all A all B set_intersection2(B,A) = set_intersection2(A,B)) # label(commutativity_k3_xboole_0) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 66 (exists A (-empty(A) & epsilon_transitive(A) & epsilon_connected(A) & ordinal(A))) # label(rc3_ordinal1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 67 (all A all B (ordinal(B) -> (in(A,B) -> ordinal(A)))) # label(t23_ordinal1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 68 (all A set_intersection2(A,empty_set) = empty_set) # label(t2_boole) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 69 (all A all B (relation(B) -> (inclusion_relation(A) = B <-> (all C all D (in(D,A) & in(C,A) -> (in(ordered_pair(C,D),B) <-> subset(C,D)))) & relation_field(B) = A))) # label(d1_wellord2) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 70 (all A all B (subset(A,B) -> A = set_intersection2(A,B))) # label(t28_xboole_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 71 (all A all B -((all C -(in(C,B) & (all D -(in(D,C) & in(D,B))))) & in(A,B))) # label(t7_tarski) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 72 (all A (relation(A) -> (is_reflexive_in(A,relation_field(A)) <-> reflexive(A)))) # label(d9_relat_2) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 73 (all A A = set_difference(A,empty_set)) # label(t3_boole) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 74 (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]. 0.96/1.29 75 (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]. 0.96/1.29 76 (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]. 0.96/1.29 77 (all A transitive(inclusion_relation(A))) # label(t3_wellord2) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 78 (all A all B all C (function(C) & relation(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]. 0.96/1.29 79 (all A all B (relation(B) & relation(A) -> relation(set_union2(A,B)))) # label(fc2_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 80 (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]. 0.96/1.29 81 (all A all B (-in(A,B) -> disjoint(singleton(A),B))) # label(l28_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 82 (all A all B -empty(ordered_pair(A,B))) # label(fc1_zfmisc_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 83 $T # label(dt_k2_tarski) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 84 (all A all B (relation(A) & relation(B) -> relation(relation_composition(A,B)))) # label(dt_k5_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 85 (all A (relation(A) -> (all B ((all C all D -(in(D,B) & -in(ordered_pair(D,C),A) & -in(ordered_pair(C,D),A) & D != C & in(C,B))) <-> is_connected_in(A,B))))) # label(d6_relat_2) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 86 (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]. 0.96/1.29 87 (all A all B (set_difference(A,singleton(B)) = A <-> -in(B,A))) # label(t65_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.29 88 (all A (function(A) & relation(A) -> (all B all C (relation_image(A,B) = C <-> (all D (in(D,C) <-> (exists E (in(E,B) & apply(A,E) = D & in(E,relation_dom(A)))))))))) # label(d12_funct_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.29 89 (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]. 0.96/1.30 90 (all A (A = union(A) <-> being_limit_ordinal(A))) # label(d6_ordinal1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 91 (all A all B (subset(singleton(A),singleton(B)) -> A = B)) # label(t6_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 92 (all A all B all C (subset(A,B) -> subset(cartesian_product2(C,A),cartesian_product2(C,B)) & subset(cartesian_product2(A,C),cartesian_product2(B,C)))) # label(t118_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 93 (all A all B subset(set_difference(A,B),A)) # label(t36_xboole_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 94 (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]. 0.96/1.30 95 (all A (relation(A) -> relation_inverse(relation_inverse(A)) = A)) # label(involutiveness_k4_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 96 (all A all B all C ((all D (in(D,C) <-> in(D,A) & in(D,B))) <-> C = set_intersection2(A,B))) # label(d3_xboole_0) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 97 (all A all B (relation(B) & function(B) -> subset(relation_image(B,relation_inverse_image(B,A)),A))) # label(t145_funct_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 98 (all A (relation(A) -> (empty_set = relation_rng(A) | empty_set = relation_dom(A) -> empty_set = A))) # label(t64_relat_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 99 (all A set_union2(A,empty_set) = A) # label(t1_boole) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 100 $T # label(dt_k3_xboole_0) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 101 (all A (A = empty_set <-> (all B -in(B,A)))) # label(d1_xboole_0) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 102 (all A all B all C (relation_of2_as_subset(C,A,B) -> ((all D -((all E -in(ordered_pair(E,D),C)) & in(D,B))) <-> relation_rng_as_subset(A,B,C) = B))) # label(t23_relset_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 103 (all A (function(A) & relation(A) -> (one_to_one(A) -> function_inverse(A) = relation_inverse(A)))) # label(d9_funct_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 104 (all A element(cast_to_subset(A),powerset(A))) # label(dt_k2_subset_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 105 (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]. 0.96/1.30 106 (all A all B (in(A,B) -> subset(A,union(B)))) # label(t92_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 107 (all A all B (relation(B) -> (well_orders(B,A) -> well_ordering(relation_restriction(B,A)) & relation_field(relation_restriction(B,A)) = A))) # label(t25_wellord2) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 108 (all A (relation(A) -> (all B all C (relation_image(A,B) = C <-> (all D (in(D,C) <-> (exists E (in(E,B) & in(ordered_pair(E,D),A))))))))) # label(d13_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 109 (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]. 0.96/1.30 110 (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]. 0.96/1.30 111 (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]. 0.96/1.30 112 (all A relation(inclusion_relation(A))) # label(dt_k1_wellord2) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 113 (all A all B A = set_intersection2(A,A)) # label(idempotence_k3_xboole_0) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 114 (all A all B (relation(B) -> (transitive(B) -> transitive(relation_restriction(B,A))))) # label(t24_wellord1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 115 (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]. 0.96/1.30 116 (all A -empty(succ(A))) # label(fc1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 117 (all A ((all B all C all D (ordinal(C) & ordinal(D) & D = B & C = B -> C = D)) -> (exists B all C (in(C,B) <-> (exists D (ordinal(C) & C = D & in(D,A))))))) # label(s1_tarski__e6_22__wellord2__1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 118 (all A ((exists B exists C A = ordered_pair(B,C)) -> (all B (pair_second(A) = B <-> (all C all D (ordered_pair(C,D) = A -> D = B)))))) # label(d2_mcart_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 119 (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]. 0.96/1.30 120 (all A all B (relation(B) -> subset(relation_image(B,A),relation_rng(B)))) # label(t144_relat_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 121 $T # label(dt_k1_setfam_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 122 (all A all B (proper_subset(A,B) -> -proper_subset(B,A))) # label(antisymmetry_r2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 123 (all A all B all C (relation(B) & function(C) & relation(C) -> (exists D ((all E all F (in(ordered_pair(apply(C,E),apply(C,F)),B) & in(F,A) & in(E,A) <-> in(ordered_pair(E,F),D))) & relation(D))))) # label(s1_relat_1__e6_21__wellord2) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 124 (exists A (relation(A) & relation_empty_yielding(A) & function(A))) # label(rc4_funct_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 125 (all A all B all C all D (subset(A,B) & subset(C,D) -> subset(cartesian_product2(A,C),cartesian_product2(B,D)))) # label(t119_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 126 (all A all B (relation(B) -> relation_rng_restriction(A,relation_dom_restriction(B,A)) = relation_restriction(B,A))) # label(t18_wellord1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 127 (all A all B (subset(A,B) <-> element(A,powerset(B)))) # label(t3_subset) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 128 (all A all B (relation(B) & function(B) -> (all C (function(C) & relation(C) -> (B = relation_dom_restriction(C,A) <-> set_intersection2(relation_dom(C),A) = relation_dom(B) & (all D (in(D,relation_dom(B)) -> apply(B,D) = apply(C,D)))))))) # label(t68_funct_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 129 (all A (relation(A) -> (all B (is_well_founded_in(A,B) <-> (all C -(empty_set != C & (all D -(disjoint(fiber(A,D),C) & in(D,C))) & subset(C,B))))))) # label(d3_wellord1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 130 (all A (ordinal(A) -> -((all B (ordinal(B) -> A != succ(B))) & -being_limit_ordinal(A)) & -((exists B (ordinal(B) & succ(B) = A)) & being_limit_ordinal(A)))) # label(t42_ordinal1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 131 (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]. 0.96/1.30 132 (all A all B all C (subset(A,C) & subset(A,B) -> subset(A,set_intersection2(B,C)))) # label(t19_xboole_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 133 (all A A = union(powerset(A))) # label(t99_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 134 (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]. 0.96/1.30 135 (all A all B all C (element(B,powerset(C)) & in(A,B) -> element(A,C))) # label(t4_subset) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 136 (all A all B all C (relation(C) & function(C) -> (in(ordered_pair(A,B),C) <-> in(A,relation_dom(C)) & apply(C,A) = B))) # label(t8_funct_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 137 (all A (empty(A) -> ordinal(A) & epsilon_connected(A) & epsilon_transitive(A))) # label(cc3_ordinal1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 138 (all A all B (equipotent(A,B) <-> are_equipotent(A,B))) # label(redefinition_r2_wellord2) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 139 (all A all B all C (relation_of2(C,A,B) -> element(relation_dom_as_subset(A,B,C),powerset(A)))) # label(dt_k4_relset_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 140 (all A (relation(A) -> (all B (relation(B) -> (all C (function(C) & relation(C) -> (relation_isomorphism(A,B,C) <-> (all D all E (in(ordered_pair(D,E),A) <-> in(ordered_pair(apply(C,D),apply(C,E)),B) & in(E,relation_field(A)) & in(D,relation_field(A)))) & one_to_one(C) & relation_field(B) = relation_rng(C) & relation_field(A) = relation_dom(C)))))))) # label(d7_wellord1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 141 (all A all B (subset(A,singleton(B)) <-> A = empty_set | A = singleton(B))) # label(t39_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 142 (all A all B (relation(A) & relation(B) -> relation(set_difference(A,B)))) # label(fc3_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 143 (exists A empty(A)) # label(rc1_xboole_0) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 144 (all A ((all B all C all D (in(ordered_pair(B,D),A) & in(ordered_pair(B,C),A) -> C = D)) <-> function(A))) # label(d1_funct_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 145 (all A all B (relation(B) & function(B) -> (in(A,relation_rng(B)) & one_to_one(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]. 0.96/1.30 146 (all A all B (ordinal(A) & ordinal(B) -> ordinal_subset(B,A) | ordinal_subset(A,B))) # label(connectedness_r1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 147 (all A all B (relation(A) -> relation(relation_dom_restriction(A,B)))) # label(dt_k7_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 148 $T # label(dt_k10_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 149 (all A all B all C (relation(C) -> (in(A,cartesian_product2(B,B)) & in(A,C) <-> in(A,relation_restriction(C,B))))) # label(t16_wellord1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 150 (all A (ordinal(A) -> ordinal(succ(A)) & epsilon_connected(succ(A)) & epsilon_transitive(succ(A)) & -empty(succ(A)))) # label(fc3_ordinal1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 151 (all A (ordinal(A) -> well_founded_relation(inclusion_relation(A)))) # label(t6_wellord2) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 152 (all A all B all C (relation(C) -> (in(A,relation_dom(C)) & in(A,B) <-> in(A,relation_dom(relation_dom_restriction(C,B)))))) # label(t86_relat_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 153 (all A (relation(A) -> (is_transitive_in(A,relation_field(A)) <-> transitive(A)))) # label(d16_relat_2) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 154 (all A (relation(A) -> relation(relation_inverse(A)))) # label(dt_k4_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 155 (all A all B all C (subset(C,cartesian_product2(A,B)) <-> relation_of2(C,A,B))) # label(d1_relset_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 156 (all A set_difference(empty_set,A) = empty_set) # label(t4_boole) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 157 (all A all B all C -(empty(C) & element(B,powerset(C)) & in(A,B))) # label(t5_subset) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 158 (all A (relation(A) -> (antisymmetric(A) <-> (all B all C (in(ordered_pair(B,C),A) & in(ordered_pair(C,B),A) -> C = B))))) # label(l3_wellord1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 159 (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)))) & A != empty_set))) # label(t32_ordinal1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 160 (all A all B (relation(B) -> subset(relation_dom_restriction(B,A),B))) # label(t88_relat_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 161 (all A set_union2(A,singleton(A)) = succ(A)) # label(d1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 162 (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]. 0.96/1.30 163 (all A all B all C -(in(B,C) & in(C,A) & in(A,B))) # label(t3_ordinal1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 164 (all A all B set_intersection2(A,B) = set_difference(A,set_difference(A,B))) # label(t48_xboole_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 165 (all A all B (in(A,B) -> -in(B,A))) # label(antisymmetry_r2_hidden) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 166 (all A ((all B (in(B,A) -> ordinal(B) & subset(B,A))) -> ordinal(A))) # label(t31_ordinal1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 167 (all A all B (relation(B) -> (identity_relation(A) = B <-> (all C all D (in(ordered_pair(C,D),B) <-> in(C,A) & C = D))))) # label(d10_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 168 (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]. 0.96/1.30 169 (all A all B all C (relation_of2(C,A,B) -> element(relation_rng_as_subset(A,B,C),powerset(B)))) # label(dt_k5_relset_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 170 (all A all B (ordinal(B) & ordinal(A) -> ordinal_subset(A,A))) # label(reflexivity_r1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 171 (all A all B (relation(B) -> (antisymmetric(B) -> antisymmetric(relation_restriction(B,A))))) # label(t25_wellord1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 172 (all A (relation(A) -> (all B ((all C ((exists D in(ordered_pair(D,C),A)) <-> in(C,B))) <-> relation_rng(A) = B)))) # label(d5_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 173 $T # label(dt_k1_mcart_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 174 (all A all B (subset(singleton(A),B) <-> in(A,B))) # label(t37_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 175 (all A all B (-empty(A) -> -empty(set_union2(B,A)))) # label(fc3_xboole_0) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 176 (all A (empty_set != A -> (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]. 0.96/1.30 177 (all A (relation(A) & function(A) -> relation(function_inverse(A)) & function(function_inverse(A)))) # label(dt_k2_funct_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 178 (all A all B all C (relation_of2(C,A,B) -> relation_rng(C) = relation_rng_as_subset(A,B,C))) # label(redefinition_k5_relset_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 179 (all A (relation(A) -> (all B (relation(B) -> (all C (function(C) & relation(C) -> (relation_isomorphism(A,B,C) & well_ordering(A) -> well_ordering(B)))))))) # label(t54_wellord1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 180 (all A (relation(A) -> (all B (relation(B) -> ((all C all D (in(ordered_pair(C,D),A) <-> in(ordered_pair(C,D),B))) <-> B = A))))) # label(d2_relat_1) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 181 (all A all B -proper_subset(A,A)) # label(irreflexivity_r2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 0.96/1.30 182 (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]. 0.96/1.30 183 (all A all B (relation(B) -> -(empty_set = relation_inverse_image(B,A) & subset(A,relation_rng(B)) & A != empty_set))) # label(t174_relat_1) # label(lemma) # label(non_clause). [assumption]. 0.96/1.30 184 (all A (relation(A) -> (all B (relation(B) -> (all C (relation(C) & function(C) -> (relation_isomorphism(A,B,C) -> (antisymmetric(A) -> antisymmetric(B)) & (well_founded_relation(A) -> well_founded_relation(B)) & (connected(A) -> connected(B)) & (transitive(A) -> transitive(B)) & (reflexive(A) -> reflexive(B))))))))) # label(t53_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.30 185 (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]. 1.01/1.30 186 (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]. 1.01/1.30 187 (all A all B all C (relation(C) -> ((exists D (in(D,relation_rng(C)) & in(ordered_pair(A,D),C) & in(D,B))) <-> in(A,relation_inverse_image(C,B))))) # label(t166_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.30 188 (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]. 1.01/1.30 189 (all A all B all C (function(C) & relation(C) -> (in(B,relation_dom(relation_dom_restriction(C,A))) -> apply(relation_dom_restriction(C,A),B) = apply(C,B)))) # label(t70_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.30 190 (all A all B set_union2(B,A) = set_union2(A,B)) # label(commutativity_k2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.01/1.30 191 (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]. 1.01/1.30 192 $T # label(dt_k1_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.30 193 (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]. 1.01/1.30 194 (all A (relation(A) -> (all B ((all C all D all E (in(C,B) & in(D,B) & in(E,B) & in(ordered_pair(D,E),A) & in(ordered_pair(C,D),A) -> in(ordered_pair(C,E),A))) <-> is_transitive_in(A,B))))) # label(d8_relat_2) # label(axiom) # label(non_clause). [assumption]. 1.01/1.30 195 (all A ((all B all C all D (in(B,A) & singleton(B) = D & C = singleton(B) & in(B,A) -> C = D)) -> (exists B (relation(B) & function(B) & (all C all D (in(ordered_pair(C,D),B) <-> singleton(C) = D & in(C,A) & in(C,A))))))) # label(s1_funct_1__e16_22__wellord2__1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.30 196 (all A all B (element(B,powerset(powerset(A))) -> (B != empty_set -> meet_of_subsets(A,complements_of_subsets(A,B)) = subset_difference(A,cast_to_subset(A),union_of_subsets(A,B))))) # label(t47_setfam_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.30 197 (all A all B ((all C (in(C,A) -> in(C,B))) <-> subset(A,B))) # label(d3_tarski) # label(axiom) # label(non_clause). [assumption]. 1.01/1.30 198 (all A (empty(A) -> function(A))) # label(cc1_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.30 199 (all A (relation(A) -> (empty_set = relation_dom(A) <-> empty_set = relation_rng(A)))) # label(t65_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.30 200 (all A all B all C all D (ordered_pair(C,D) = ordered_pair(A,B) -> B = D & A = C)) # label(t33_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.30 201 (all A (ordinal(A) -> well_ordering(inclusion_relation(A)))) # label(t7_wellord2) # label(lemma) # label(non_clause). [assumption]. 1.01/1.30 202 $T # label(dt_m1_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.30 203 (all A ((exists B exists C ordered_pair(B,C) = A) -> (all B (pair_first(A) = B <-> (all C all D (A = ordered_pair(C,D) -> B = C)))))) # label(d1_mcart_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.30 204 (all A (relation(A) & function(A) & one_to_one(A) -> relation(relation_inverse(A)) & function(relation_inverse(A)))) # label(fc3_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.30 205 (all A all B all C (relation(B) & relation(C) & function(C) -> ((all D all E all F (E = D & F = D & (exists I exists J (in(ordered_pair(apply(C,I),apply(C,J)),B) & F = ordered_pair(I,J))) & (exists G exists H (E = ordered_pair(G,H) & in(ordered_pair(apply(C,G),apply(C,H)),B))) -> E = F)) -> (exists D all E (in(E,D) <-> (exists F (in(F,cartesian_product2(A,A)) & (exists K exists L (in(ordered_pair(apply(C,K),apply(C,L)),B) & ordered_pair(K,L) = E)) & F = E))))))) # label(s1_tarski__e6_21__wellord2__1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 206 (all A all B all C (unordered_pair(B,C) = singleton(A) -> A = B)) # label(t8_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 207 (all A all B -(subset(A,B) & proper_subset(B,A))) # label(t60_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 208 (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]. 1.01/1.31 209 (exists A -empty(A)) # label(rc2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 210 (exists A (relation(A) & -empty(A))) # label(rc2_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 211 (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]. 1.01/1.31 212 (all A all B (relation(B) & relation(A) -> relation(set_intersection2(A,B)))) # label(fc1_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 213 (all A (function(A) & relation(A) -> ((all B all C (apply(A,C) = apply(A,B) & in(C,relation_dom(A)) & in(B,relation_dom(A)) -> C = B)) <-> one_to_one(A)))) # label(d8_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 214 (all A all B -(A != B & empty(B) & empty(A))) # label(t8_boole) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 215 (all A all B all C (relation(C) -> (in(ordered_pair(A,B),C) -> in(A,relation_field(C)) & in(B,relation_field(C))))) # label(t30_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 216 (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]. 1.01/1.31 217 (all A (relation(A) & empty(A) & function(A) -> relation(A) & function(A) & one_to_one(A))) # label(cc2_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 218 (all A (relation(A) -> (all B ((all C (in(C,B) <-> (exists D in(ordered_pair(C,D),A)))) <-> relation_dom(A) = B)))) # label(d4_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 219 (all A (relation(A) -> ((all B all C -in(ordered_pair(B,C),A)) -> empty_set = A))) # label(t56_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 220 (all A all B equipotent(A,A)) # label(reflexivity_r2_wellord2) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 221 (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]. 1.01/1.31 222 (all A (ordinal(A) -> epsilon_transitive(A) & epsilon_connected(A))) # label(cc1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 223 (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]. 1.01/1.31 224 (all A all B (element(B,powerset(A)) -> subset_complement(A,subset_complement(A,B)) = B)) # label(involutiveness_k3_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 225 $T # label(dt_k3_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 226 (all A all B (in(A,B) -> subset(A,union(B)))) # label(l50_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 227 (all A (epsilon_transitive(A) & epsilon_connected(A) -> ordinal(A))) # label(cc2_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 228 (all A all B A = set_union2(A,A)) # label(idempotence_k2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 229 (all A (relation(A) -> (all B (well_orders(A,B) <-> is_reflexive_in(A,B) & is_well_founded_in(A,B) & is_connected_in(A,B) & is_antisymmetric_in(A,B) & is_transitive_in(A,B))))) # label(d5_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 230 (all A all B (element(B,powerset(powerset(A))) -> meet_of_subsets(A,B) = set_meet(B))) # label(redefinition_k6_setfam_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 231 (all A (relation(A) -> (all B (relation(B) -> (subset(relation_rng(A),relation_dom(B)) -> relation_dom(relation_composition(A,B)) = relation_dom(A)))))) # label(t46_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 232 (all A all B (function(B) & relation(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]. 1.01/1.31 233 (all A all B ((all C all D all E ((exists H exists I (ordered_pair(H,I) = E & in(H,A) & singleton(H) = I)) & E = C & (exists F exists G (D = ordered_pair(F,G) & in(F,A) & singleton(F) = G)) & D = C -> D = E)) -> (exists C all D (in(D,C) <-> (exists E ((exists J exists K (K = singleton(J) & in(J,A) & D = ordered_pair(J,K))) & E = D & in(E,cartesian_product2(A,B)))))))) # label(s1_tarski__e16_22__wellord2__2) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 234 (all A all B ((all C (C = A <-> in(C,B))) <-> B = singleton(A))) # label(d1_tarski) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 235 (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]. 1.01/1.31 236 (all A all B exists C relation_of2(C,A,B)) # label(existence_m1_relset_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 237 (all A all B unordered_pair(B,A) = unordered_pair(A,B)) # label(commutativity_k2_tarski) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 238 (all A all B set_difference(set_union2(A,B),B) = set_difference(A,B)) # label(t40_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 239 (all A all B (B = A <-> subset(A,B) & subset(B,A))) # label(d10_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 240 (all A (ordinal(A) -> (all B (ordinal(B) -> (ordinal_subset(succ(A),B) <-> in(A,B)))))) # label(t33_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 241 (exists A (relation(A) & relation_empty_yielding(A))) # label(rc3_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 242 (all A (relation(A) & function(A) -> (one_to_one(A) -> relation_dom(A) = relation_rng(function_inverse(A)) & relation_dom(function_inverse(A)) = relation_rng(A)))) # label(t55_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 243 (all A (epsilon_transitive(A) <-> (all B (in(B,A) -> subset(B,A))))) # label(d2_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 244 (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]. 1.01/1.31 245 (all A (subset(A,empty_set) -> empty_set = A)) # label(t3_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 246 $T # label(dt_k1_zfmisc_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 247 (all A all B (set_difference(A,B) = A <-> disjoint(A,B))) # label(t83_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 248 (all A (relation(A) & function(A) -> (all B all C ((-in(B,relation_dom(A)) -> (C = empty_set <-> C = apply(A,B))) & (in(B,relation_dom(A)) -> (in(ordered_pair(B,C),A) <-> apply(A,B) = C)))))) # label(d4_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 249 (all A (relation(A) -> (is_antisymmetric_in(A,relation_field(A)) <-> antisymmetric(A)))) # label(d12_relat_2) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 250 (all A -empty(powerset(A))) # label(fc1_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 251 $T # label(dt_k2_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 252 (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]. 1.01/1.31 253 (all A all B all C (relation(C) -> ((exists D (in(D,relation_dom(C)) & in(D,B) & in(ordered_pair(D,A),C))) <-> in(A,relation_image(C,B))))) # label(t143_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 254 (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]. 1.01/1.31 255 (all A all B (element(B,powerset(powerset(A))) -> (all C (element(C,powerset(powerset(A))) -> ((all D (element(D,powerset(A)) -> (in(subset_complement(A,D),B) <-> in(D,C)))) <-> complements_of_subsets(A,B) = C))))) # label(d8_setfam_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 256 (all A exists B (empty(B) & element(B,powerset(A)))) # label(rc2_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 257 (all A ((all B all C all D (in(B,A) & D = singleton(B) & singleton(B) = C & in(B,A) -> D = C)) -> (exists B all C (in(C,B) <-> (exists D (C = singleton(D) & in(D,A) & in(D,A))))))) # label(s1_tarski__e16_22__wellord2__1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 258 (all A (function(A) & relation(A) -> (one_to_one(A) -> one_to_one(function_inverse(A))))) # label(t62_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 259 (all A (relation(A) -> (connected(A) <-> is_connected_in(A,relation_field(A))))) # label(d14_relat_2) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 260 (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]. 1.01/1.31 261 (all A (-empty(A) & relation(A) -> -empty(relation_rng(A)))) # label(fc6_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 262 (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]. 1.01/1.31 263 (all A A = cast_to_subset(A)) # label(d4_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 264 $T # label(dt_k2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 265 (all A subset(empty_set,A)) # label(t2_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 266 (all A all B all C (relation(B) & relation(C) & function(C) -> (exists D all E (in(E,D) <-> in(E,cartesian_product2(A,A)) & (exists F exists G (in(ordered_pair(apply(C,F),apply(C,G)),B) & E = ordered_pair(F,G))))))) # label(s1_xboole_0__e6_21__wellord2__1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 267 (all A all B ((all C (in(C,A) <-> in(C,B))) -> B = A)) # label(t2_tarski) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 268 (all A all B (relation(B) -> (connected(B) -> connected(relation_restriction(B,A))))) # label(t23_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 269 (all A all B ((A != empty_set -> ((all C (in(C,B) <-> (all D (in(D,A) -> in(C,D))))) <-> set_meet(A) = B)) & (A = empty_set -> (empty_set = B <-> B = set_meet(A))))) # label(d1_setfam_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 270 (all A all B ((empty(A) -> (element(B,A) <-> empty(B))) & (-empty(A) -> (in(B,A) <-> element(B,A))))) # label(d2_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 271 (all A (ordinal(A) -> (all B (ordinal(B) -> -(-in(B,A) & B != A & -in(A,B)))))) # label(t24_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 272 (all A (ordinal(A) <-> epsilon_connected(A) & epsilon_transitive(A))) # label(d4_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 273 (all A all B (equipotent(A,B) -> equipotent(B,A))) # label(symmetry_r2_wellord2) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 274 (exists A (epsilon_transitive(A) & ordinal(A) & epsilon_connected(A))) # label(rc1_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 275 (all A all B (subset(A,B) -> B = set_union2(A,B))) # label(t12_xboole_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 276 (all A (empty(A) -> relation(A))) # label(cc1_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 277 (all A in(A,succ(A))) # label(t10_ordinal1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 278 (all A all B all C (relation(C) & function(C) -> (in(B,A) -> apply(relation_dom_restriction(C,A),B) = apply(C,B)))) # label(t72_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 279 (all A (relation(A) & -empty(A) -> -empty(relation_dom(A)))) # label(fc5_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 280 (all A all B (empty_set = set_intersection2(A,B) <-> disjoint(A,B))) # label(d7_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 281 (all A (relation(A) -> (all B (relation(B) -> (subset(relation_dom(A),relation_rng(B)) -> relation_rng(A) = relation_rng(relation_composition(B,A))))))) # label(t47_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 282 (all A all B (in(B,A) -> apply(identity_relation(A),B) = B)) # label(t35_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 283 (all A all B exists C all D (in(D,cartesian_product2(A,B)) & (exists E exists F (F = singleton(E) & in(E,A) & ordered_pair(E,F) = D)) <-> in(D,C))) # label(s1_xboole_0__e16_22__wellord2__1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 284 $T # label(dt_k2_zfmisc_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 285 (all A all B (element(A,B) -> in(A,B) | empty(B))) # label(t2_subset) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 286 (all A all B (in(A,B) -> B = set_union2(singleton(A),B))) # label(l23_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 287 (all A all B (in(A,B) -> set_union2(singleton(A),B) = B)) # label(t46_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 288 (all A all B (function(B) & relation(B) -> (all C (relation(C) & function(C) -> (in(A,relation_dom(C)) & in(apply(C,A),relation_dom(B)) <-> in(A,relation_dom(relation_composition(C,B)))))))) # label(t21_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 289 (all A (relation(A) -> subset(A,cartesian_product2(relation_dom(A),relation_rng(A))))) # label(t21_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 290 (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]. 1.01/1.31 291 (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]. 1.01/1.31 292 (all A (relation(A) -> (all B (relation(B) -> (all C (relation(C) -> ((all D all E ((exists F (in(ordered_pair(F,E),B) & in(ordered_pair(D,F),A))) <-> in(ordered_pair(D,E),C))) <-> relation_composition(A,B) = C))))))) # label(d8_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 293 (all A all B (relation(B) -> (well_ordering(B) -> well_ordering(relation_restriction(B,A))))) # label(t32_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 294 (all A all B (-empty(A) -> -empty(set_union2(A,B)))) # label(fc2_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 295 (all A all B (-((exists C in(C,set_intersection2(A,B))) & disjoint(A,B)) & -(-disjoint(A,B) & (all C -in(C,set_intersection2(A,B)))))) # label(t4_xboole_0) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 296 (all A ((all B all C -(in(B,A) & -in(B,C) & C != B & -in(C,B) & in(C,A))) <-> epsilon_connected(A))) # label(d3_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 297 (all A all B all C (singleton(A) = unordered_pair(B,C) -> B = C)) # label(t9_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 298 (all A (relation(A) -> (well_ordering(A) <-> well_orders(A,relation_field(A))))) # label(t8_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 299 (all A reflexive(inclusion_relation(A))) # label(t2_wellord2) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 300 (all A all B (relation(B) -> relation(relation_rng_restriction(A,B)))) # label(dt_k8_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 301 $T # label(dt_k4_tarski) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 302 (all A exists B (in(A,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) & -are_equipotent(C,B) & -in(C,B))))) # label(t9_tarski) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 303 (all A (relation(A) -> (well_founded_relation(A) <-> is_well_founded_in(A,relation_field(A))))) # label(t5_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 304 (all A all B (-empty(B) & -empty(A) -> -empty(cartesian_product2(A,B)))) # label(fc4_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 305 $T # label(dt_k1_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 306 (all A all B (relation(B) & function(B) -> (all C (function(C) & relation(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]. 1.01/1.31 307 (all A all B -(empty(B) & in(A,B))) # label(t7_boole) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 308 $T # label(dt_k1_enumset1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 309 (all A (function(identity_relation(A)) & relation(identity_relation(A)))) # label(fc2_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 310 (all A (function(A) & relation(A) -> (one_to_one(A) -> (all B (relation(B) & function(B) -> (relation_rng(A) = relation_dom(B) & (all C all D ((in(C,relation_rng(A)) & D = apply(B,C) -> C = apply(A,D) & in(D,relation_dom(A))) & (C = apply(A,D) & in(D,relation_dom(A)) -> apply(B,C) = D & in(C,relation_rng(A))))) <-> function_inverse(A) = B)))))) # label(t54_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 311 (all A ((all B all C all D (in(B,A) & singleton(B) = D & C = singleton(B) -> C = D)) & (all B -((all C singleton(B) != C) & in(B,A))) -> (exists B (relation(B) & function(B) & (all C (in(C,A) -> singleton(C) = apply(B,C))) & A = relation_dom(B))))) # label(s2_funct_1__e16_22__wellord2__1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 312 (all A all B all C (element(B,powerset(A)) & element(C,powerset(A)) -> subset_difference(A,B,C) = set_difference(B,C))) # label(redefinition_k6_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 313 (all A antisymmetric(inclusion_relation(A))) # label(t5_wellord2) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 314 (all A all B all C (C = set_difference(A,B) <-> (all D (in(D,A) & -in(D,B) <-> in(D,C))))) # label(d4_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 315 (all A all B (proper_subset(A,B) <-> A != B & subset(A,B))) # label(d8_xboole_0) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 316 (all A all B (relation(B) -> subset(relation_field(relation_restriction(B,A)),relation_field(B)) & subset(relation_field(relation_restriction(B,A)),A))) # label(t20_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 317 (all A all B (relation(A) -> relation(relation_restriction(A,B)))) # label(dt_k2_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 318 (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]. 1.01/1.31 319 (all A ((exists B (in(B,A) & ordinal(B))) -> (exists B ((all C (ordinal(C) -> (in(C,A) -> ordinal_subset(B,C)))) & in(B,A) & ordinal(B))))) # label(s1_ordinal1__e8_6__wellord2) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 320 (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]. 1.01/1.31 321 (all A all B (pair_first(ordered_pair(A,B)) = A & pair_second(ordered_pair(A,B)) = B)) # label(t7_mcart_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 322 (exists A (relation(A) & empty(A) & function(A))) # label(rc2_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 323 (all A all B (powerset(A) = B <-> (all C (in(C,B) <-> subset(C,A))))) # label(d1_zfmisc_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.31 324 (all A (A = relation_rng(identity_relation(A)) & relation_dom(identity_relation(A)) = A)) # label(t71_relat_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.31 325 (all A all B (union(A) = B <-> (all C ((exists D (in(D,A) & in(C,D))) <-> in(C,B))))) # label(d4_tarski) # label(axiom) # label(non_clause). [assumption]. 1.01/1.32 326 (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]. 1.01/1.32 327 (all A all B (ordinal(B) -> (exists C all D (in(D,succ(B)) & (exists E (in(E,A) & E = D & ordinal(E))) <-> in(D,C))))) # label(s1_xboole_0__e8_6__wellord2__1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.32 328 (all A all B (function(A) & function(B) & relation(B) & relation(A) -> relation(relation_composition(A,B)) & function(relation_composition(A,B)))) # label(fc1_funct_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.32 329 (all A -empty(singleton(A))) # label(fc2_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.32 330 (all A (relation(A) -> (well_ordering(A) <-> reflexive(A) & transitive(A) & antisymmetric(A) & connected(A) & well_founded_relation(A)))) # label(d4_wellord1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.32 331 (all A all B (relation(B) & function(B) -> (A = relation_dom(B) & (all C (in(C,A) -> apply(B,C) = C)) <-> identity_relation(A) = B))) # label(t34_funct_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.32 332 (all A exists B (in(A,B) & (all C -(-are_equipotent(C,B) & -in(C,B) & subset(C,B))) & (all C (in(C,B) -> in(powerset(C),B))) & (all C all D (subset(D,C) & in(C,B) -> in(D,B))))) # label(t136_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.32 333 (all A all B (-(disjoint(A,B) & (exists C (in(C,B) & in(C,A)))) & -((all C -(in(C,A) & in(C,B))) & -disjoint(A,B)))) # label(t3_xboole_0) # label(lemma) # label(non_clause). [assumption]. 1.01/1.32 334 (all A all B all C all D (in(A,C) & in(B,D) <-> in(ordered_pair(A,B),cartesian_product2(C,D)))) # label(t106_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.32 335 (all A all B unordered_pair(unordered_pair(A,B),singleton(A)) = ordered_pair(A,B)) # label(d5_tarski) # label(axiom) # label(non_clause). [assumption]. 1.01/1.32 336 (all A all B ((exists C (relation(C) & function(C) & one_to_one(C) & relation_dom(C) = A & B = relation_rng(C))) <-> equipotent(A,B))) # label(d4_wellord2) # label(axiom) # label(non_clause). [assumption]. 1.01/1.32 337 (all A (relation(A) -> (all B all C (C = relation_inverse_image(A,B) <-> (all D ((exists E (in(E,B) & in(ordered_pair(D,E),A))) <-> in(D,C))))))) # label(d14_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.32 338 (all A all B all C (unordered_pair(A,B) = C <-> (all D (D = B | A = D <-> in(D,C))))) # label(d2_tarski) # label(axiom) # label(non_clause). [assumption]. 1.01/1.32 339 (all A all B (in(A,B) -> element(A,B))) # label(t1_subset) # label(axiom) # label(non_clause). [assumption]. 1.01/1.32 340 (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]. 1.01/1.32 341 (exists A (relation(A) & empty(A))) # label(rc1_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.32 342 (exists A (one_to_one(A) & empty(A) & ordinal(A) & epsilon_connected(A) & epsilon_transitive(A) & function(A) & relation(A))) # label(rc2_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.32 343 (all A all B all C all D (relation_of2_as_subset(D,C,A) -> (subset(A,B) -> relation_of2_as_subset(D,C,B)))) # label(t16_relset_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.32 344 (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]. 1.01/1.32 345 (all A all B (relation(B) -> (reflexive(B) -> reflexive(relation_restriction(B,A))))) # label(t22_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.32 346 (all A all B (element(B,powerset(powerset(A))) -> -(empty_set = complements_of_subsets(A,B) & empty_set != B))) # label(t46_setfam_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.32 347 (all A relation(identity_relation(A))) # label(dt_k6_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.33 348 (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]. 1.01/1.33 349 (all A all B (relation(B) -> (subset(A,relation_field(B)) & well_ordering(B) -> A = relation_field(relation_restriction(B,A))))) # label(t39_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.33 350 (all A (relation(A) -> ((all B all C -(in(B,relation_field(A)) & in(C,relation_field(A)) & -in(ordered_pair(C,B),A) & -in(ordered_pair(B,C),A) & C != B)) <-> connected(A)))) # label(l4_wellord1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.33 351 (all A all B -(in(A,B) & disjoint(singleton(A),B))) # label(l25_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.33 352 $T # label(dt_k2_mcart_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.33 353 (all A (relation(A) -> (all B (relation(B) -> (relation_inverse(A) = B <-> (all C all D (in(ordered_pair(C,D),B) <-> in(ordered_pair(D,C),A)))))))) # label(d7_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.33 354 (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]. 1.01/1.33 355 (all A all B (relation(B) & empty(A) -> relation(relation_composition(B,A)) & empty(relation_composition(B,A)))) # label(fc10_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.33 356 (all A (ordinal(A) -> ordinal(union(A)) & epsilon_connected(union(A)) & epsilon_transitive(union(A)))) # label(fc4_ordinal1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.33 357 (all A (relation(A) -> (all B ((all C all D (in(C,B) & in(D,B) & in(ordered_pair(C,D),A) & in(ordered_pair(D,C),A) -> D = C)) <-> is_antisymmetric_in(A,B))))) # label(d4_relat_2) # label(axiom) # label(non_clause). [assumption]. 1.01/1.33 358 (all A (empty(A) -> relation(relation_dom(A)) & empty(relation_dom(A)))) # label(fc7_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.33 359 (all A (-empty(A) -> (exists B (-empty(B) & element(B,powerset(A)))))) # label(rc1_subset_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.33 360 $T # label(dt_k3_tarski) # label(axiom) # label(non_clause). [assumption]. 1.01/1.33 361 (all A (empty(A) -> empty(relation_rng(A)) & relation(relation_rng(A)))) # label(fc8_relat_1) # label(axiom) # label(non_clause). [assumption]. 1.01/1.33 362 (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]. 1.01/1.33 363 (all A all B (subset(singleton(A),B) <-> in(A,B))) # label(l2_zfmisc_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.33 364 (all A all B (element(B,powerset(powerset(A))) -> (empty_set != B -> union_of_subsets(A,complements_of_subsets(A,B)) = subset_difference(A,cast_to_subset(A),meet_of_subsets(A,B))))) # label(t48_setfam_1) # label(lemma) # label(non_clause). [assumption]. 1.01/1.33 365 -(all A exists B (function(B) & relation_dom(B) = A & (all C (in(C,A) -> apply(B,C) = singleton(C))) & relation(B))) # label(s3_funct_1__e16_22__wellord2) # label(negated_conjecture) # label(non_clause). [assumption]. 1.01/1.33 1.01/1.33 ============================== end of process non-clausal formulas === 1.01/1.33 1.01/1.33 ============================== PROCESS INITIAL CLAUSES =============== 1.01/1.33 1.01/1.33 ============================== PREDICATE ELIMINATION ================= 1.01/1.33 366 relation_of2_as_subset(A,B,C) | -relation_of2(A,B,C) # label(redefinition_m2_relset_1) # label(axiom). [clausify(15)]. 1.01/1.33 367 -relation_of2_as_subset(A,B,C) | relation_of2(A,B,C) # label(redefinition_m2_relset_1) # label(axiom). [clausify(15)]. 1.01/1.33 368 -relation_of2(A,B,C) | relation_dom_as_subset(B,C,A) = relation_dom(A) # label(redefinition_k4_relset_1) # label(axiom). [clausify(52)]. 1.01/1.33 Derived: relation_dom_as_subset(A,B,C) = relation_dom(C) | -relation_of2_as_subset(C,A,B). [resolve(368,a,367,b)]. 1.01/1.33 369 -relation_of2(A,B,C) | element(relation_dom_as_subset(B,C,A),powerset(B)) # label(dt_k4_relset_1) # label(axiom). [clausify(139)]. 1.01/1.33 Derived: element(relation_dom_as_subset(A,B,C),powerset(A)) | -relation_of2_as_subset(C,A,B). [resolve(369,a,367,b)]. 1.01/1.42 370 -subset(A,cartesian_product2(B,C)) | relation_of2(A,B,C) # label(d1_relset_1) # label(axiom). [clausify(155)]. 1.01/1.42 Derived: -subset(A,cartesian_product2(B,C)) | relation_of2_as_subset(A,B,C). [resolve(370,b,366,b)]. 1.01/1.42 Derived: -subset(A,cartesian_product2(B,C)) | relation_dom_as_subset(B,C,A) = relation_dom(A). [resolve(370,b,368,a)]. 1.01/1.42 Derived: -subset(A,cartesian_product2(B,C)) | element(relation_dom_as_subset(B,C,A),powerset(B)). [resolve(370,b,369,a)]. 1.01/1.42 371 subset(A,cartesian_product2(B,C)) | -relation_of2(A,B,C) # label(d1_relset_1) # label(axiom). [clausify(155)]. 1.01/1.42 Derived: subset(A,cartesian_product2(B,C)) | -relation_of2_as_subset(A,B,C). [resolve(371,b,367,b)]. 1.01/1.42 372 -relation_of2(A,B,C) | element(relation_rng_as_subset(B,C,A),powerset(C)) # label(dt_k5_relset_1) # label(axiom). [clausify(169)]. 1.01/1.42 Derived: element(relation_rng_as_subset(A,B,C),powerset(B)) | -relation_of2_as_subset(C,A,B). [resolve(372,a,367,b)]. 1.01/1.42 Derived: element(relation_rng_as_subset(A,B,C),powerset(B)) | -subset(C,cartesian_product2(A,B)). [resolve(372,a,370,b)]. 1.01/1.42 373 -relation_of2(A,B,C) | relation_rng_as_subset(B,C,A) = relation_rng(A) # label(redefinition_k5_relset_1) # label(axiom). [clausify(178)]. 1.01/1.42 Derived: relation_rng_as_subset(A,B,C) = relation_rng(C) | -relation_of2_as_subset(C,A,B). [resolve(373,a,367,b)]. 1.01/1.42 Derived: relation_rng_as_subset(A,B,C) = relation_rng(C) | -subset(C,cartesian_product2(A,B)). [resolve(373,a,370,b)]. 1.01/1.42 374 relation_of2(f126(A,B),A,B) # label(existence_m1_relset_1) # label(axiom). [clausify(236)]. 1.01/1.42 Derived: relation_of2_as_subset(f126(A,B),A,B). [resolve(374,a,366,b)]. 1.01/1.42 Derived: relation_dom_as_subset(A,B,f126(A,B)) = relation_dom(f126(A,B)). [resolve(374,a,368,a)]. 1.01/1.42 Derived: element(relation_dom_as_subset(A,B,f126(A,B)),powerset(A)). [resolve(374,a,369,a)]. 1.01/1.42 Derived: subset(f126(A,B),cartesian_product2(A,B)). [resolve(374,a,371,b)]. 1.01/1.42 Derived: element(relation_rng_as_subset(A,B,f126(A,B)),powerset(B)). [resolve(374,a,372,a)]. 1.01/1.42 Derived: relation_rng_as_subset(A,B,f126(A,B)) = relation_rng(f126(A,B)). [resolve(374,a,373,a)]. 1.01/1.42 375 -ordinal(A) | being_limit_ordinal(A) | ordinal(f24(A)) # label(t41_ordinal1) # label(lemma). [clausify(27)]. 1.01/1.42 376 -ordinal(A) | -being_limit_ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A) # label(t41_ordinal1) # label(lemma). [clausify(27)]. 1.01/1.42 Derived: -ordinal(A) | ordinal(f24(A)) | -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A). [resolve(375,b,376,b)]. 1.01/1.42 377 -ordinal(A) | being_limit_ordinal(A) | in(f24(A),A) # label(t41_ordinal1) # label(lemma). [clausify(27)]. 1.01/1.42 Derived: -ordinal(A) | in(f24(A),A) | -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A). [resolve(377,b,376,b)]. 1.01/1.42 378 -ordinal(A) | being_limit_ordinal(A) | -in(succ(f24(A)),A) # label(t41_ordinal1) # label(lemma). [clausify(27)]. 1.01/1.42 Derived: -ordinal(A) | -in(succ(f24(A)),A) | -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A). [resolve(378,b,376,b)]. 1.01/1.42 379 union(A) != A | being_limit_ordinal(A) # label(d6_ordinal1) # label(axiom). [clausify(90)]. 1.01/1.42 Derived: union(A) != A | -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A). [resolve(379,b,376,b)]. 1.01/1.42 380 union(A) = A | -being_limit_ordinal(A) # label(d6_ordinal1) # label(axiom). [clausify(90)]. 1.01/1.42 Derived: union(A) = A | -ordinal(A) | ordinal(f24(A)). [resolve(380,b,375,b)]. 1.01/1.42 Derived: union(A) = A | -ordinal(A) | in(f24(A),A). [resolve(380,b,377,b)]. 1.01/1.42 Derived: union(A) = A | -ordinal(A) | -in(succ(f24(A)),A). [resolve(380,b,378,b)]. 1.01/1.42 381 -ordinal(A) | ordinal(f63(A)) | being_limit_ordinal(A) # label(t42_ordinal1) # label(lemma). [clausify(130)]. 1.01/1.42 Derived: -ordinal(A) | ordinal(f63(A)) | -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A). [resolve(381,c,376,b)]. 1.01/1.42 Derived: -ordinal(A) | ordinal(f63(A)) | union(A) = A. [resolve(381,c,380,b)]. 1.01/1.42 382 -ordinal(A) | succ(f63(A)) = A | being_limit_ordinal(A) # label(t42_ordinal1) # label(lemma). [clausify(130)]. 1.01/1.42 Derived: -ordinal(A) | succ(f63(A)) = A | -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A). [resolve(382,c,376,b)]. 1.01/1.42 Derived: -ordinal(A) | succ(f63(A)) = A | union(A) = A. [resolve(382,c,380,b)]. 1.01/1.52 383 -ordinal(A) | -ordinal(B) | succ(B) != A | -being_limit_ordinal(A) # label(t42_ordinal1) # label(lemma). [clausify(130)]. 1.01/1.52 Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | -ordinal(A) | ordinal(f24(A)). [resolve(383,d,375,b)]. 1.01/1.52 Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | -ordinal(A) | in(f24(A),A). [resolve(383,d,377,b)]. 1.01/1.52 Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | -ordinal(A) | -in(succ(f24(A)),A). [resolve(383,d,378,b)]. 1.01/1.52 Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | union(A) != A. [resolve(383,d,379,b)]. 1.01/1.52 Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | -ordinal(A) | ordinal(f63(A)). [resolve(383,d,381,c)]. 1.01/1.52 Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | -ordinal(A) | succ(f63(A)) = A. [resolve(383,d,382,c)]. 1.01/1.52 384 -epsilon_transitive(A) | -ordinal(B) | -proper_subset(A,B) | in(A,B) # label(t21_ordinal1) # label(lemma). [clausify(86)]. 1.01/1.52 385 epsilon_transitive(c3) # label(rc3_ordinal1) # label(axiom). [clausify(66)]. 1.01/1.52 Derived: -ordinal(A) | -proper_subset(c3,A) | in(c3,A). [resolve(384,a,385,a)]. 1.01/1.52 386 -empty(A) | epsilon_transitive(A) # label(cc3_ordinal1) # label(axiom). [clausify(137)]. 1.01/1.52 Derived: -empty(A) | -ordinal(B) | -proper_subset(A,B) | in(A,B). [resolve(386,b,384,a)]. 1.01/1.52 387 epsilon_transitive(empty_set) # label(fc2_ordinal1_AndRHS_AndRHS_AndRHS_AndRHS_AndRHS_AndRHS_AndLHS) # label(axiom). [assumption]. 1.01/1.52 Derived: -ordinal(A) | -proper_subset(empty_set,A) | in(empty_set,A). [resolve(387,a,384,a)]. 1.01/1.52 388 -ordinal(A) | epsilon_transitive(succ(A)) # label(fc3_ordinal1) # label(axiom). [clausify(150)]. 1.01/1.52 Derived: -ordinal(A) | -ordinal(B) | -proper_subset(succ(A),B) | in(succ(A),B). [resolve(388,b,384,a)]. 1.01/1.52 389 -ordinal(A) | epsilon_transitive(A) # label(cc1_ordinal1) # label(axiom). [clausify(222)]. 1.01/1.52 Derived: -ordinal(A) | -ordinal(B) | -proper_subset(A,B) | in(A,B). [resolve(389,b,384,a)]. 1.01/1.52 390 -epsilon_transitive(A) | -epsilon_connected(A) | ordinal(A) # label(cc2_ordinal1) # label(axiom). [clausify(227)]. 1.01/1.52 Derived: -epsilon_connected(c3) | ordinal(c3). [resolve(390,a,385,a)]. 1.01/1.52 Derived: -epsilon_connected(A) | ordinal(A) | -empty(A). [resolve(390,a,386,b)]. 1.01/1.52 Derived: -epsilon_connected(empty_set) | ordinal(empty_set). [resolve(390,a,387,a)]. 1.01/1.52 Derived: -epsilon_connected(succ(A)) | ordinal(succ(A)) | -ordinal(A). [resolve(390,a,388,b)]. 1.01/1.52 391 -epsilon_transitive(A) | -in(B,A) | subset(B,A) # label(d2_ordinal1) # label(axiom). [clausify(243)]. 1.01/1.52 Derived: -in(A,c3) | subset(A,c3). [resolve(391,a,385,a)]. 1.01/1.52 Derived: -in(A,empty_set) | subset(A,empty_set). [resolve(391,a,387,a)]. 1.01/1.52 Derived: -in(A,succ(B)) | subset(A,succ(B)) | -ordinal(B). [resolve(391,a,388,b)]. 1.01/1.52 Derived: -in(A,B) | subset(A,B) | -ordinal(B). [resolve(391,a,389,b)]. 1.01/1.52 392 epsilon_transitive(A) | in(f127(A),A) # label(d2_ordinal1) # label(axiom). [clausify(243)]. 1.01/1.52 Derived: in(f127(A),A) | -ordinal(B) | -proper_subset(A,B) | in(A,B). [resolve(392,a,384,a)]. 1.01/1.52 Derived: in(f127(A),A) | -epsilon_connected(A) | ordinal(A). [resolve(392,a,390,a)]. 1.01/1.52 Derived: in(f127(A),A) | -in(B,A) | subset(B,A). [resolve(392,a,391,a)]. 1.01/1.52 393 epsilon_transitive(A) | -subset(f127(A),A) # label(d2_ordinal1) # label(axiom). [clausify(243)]. 1.01/1.52 Derived: -subset(f127(A),A) | -ordinal(B) | -proper_subset(A,B) | in(A,B). [resolve(393,a,384,a)]. 1.01/1.52 Derived: -subset(f127(A),A) | -epsilon_connected(A) | ordinal(A). [resolve(393,a,390,a)]. 1.01/1.52 Derived: -subset(f127(A),A) | -in(B,A) | subset(B,A). [resolve(393,a,391,a)]. 1.01/1.52 394 -ordinal(A) | epsilon_transitive(A) # label(d4_ordinal1) # label(axiom). [clausify(272)]. 1.01/1.52 395 ordinal(A) | -epsilon_connected(A) | -epsilon_transitive(A) # label(d4_ordinal1) # label(axiom). [clausify(272)]. 1.01/1.52 396 epsilon_transitive(c9) # label(rc1_ordinal1) # label(axiom). [clausify(274)]. 1.01/1.52 Derived: -ordinal(A) | -proper_subset(c9,A) | in(c9,A). [resolve(396,a,384,a)]. 1.01/1.52 Derived: -in(A,c9) | subset(A,c9). [resolve(396,a,391,a)]. 1.01/1.52 397 epsilon_transitive(c12) # label(rc2_ordinal1) # label(axiom). [clausify(342)]. 1.01/1.52 Derived: -ordinal(A) | -proper_subset(c12,A) | in(c12,A). [resolve(397,a,384,a)]. 1.01/1.52 Derived: -epsilon_connected(c12) | ordinal(c12). [resolve(397,a,390,a)]. 2.34/2.68 Derived: -in(A,c12) | subset(A,c12). [resolve(397,a,391,a)]. 2.34/2.68 398 -ordinal(A) | epsilon_transitive(union(A)) # label(fc4_ordinal1) # label(axiom). [clausify(356)]. 2.34/2.68 Derived: -ordinal(A) | -ordinal(B) | -proper_subset(union(A),B) | in(union(A),B). [resolve(398,b,384,a)]. 2.34/2.68 Derived: -ordinal(A) | -epsilon_connected(union(A)) | ordinal(union(A)). [resolve(398,b,390,a)]. 2.34/2.68 Derived: -ordinal(A) | -in(B,union(A)) | subset(B,union(A)). [resolve(398,b,391,a)]. 2.34/2.68 399 equipotent(A,B) | -are_equipotent(A,B) # label(redefinition_r2_wellord2) # label(axiom). [clausify(138)]. 2.34/2.68 400 -equipotent(A,B) | are_equipotent(A,B) # label(redefinition_r2_wellord2) # label(axiom). [clausify(138)]. 2.34/2.68 401 -subset(A,f153(B)) | are_equipotent(A,f153(B)) | in(A,f153(B)) # label(t9_tarski) # label(axiom). [clausify(302)]. 2.34/2.68 Derived: -subset(A,f153(B)) | in(A,f153(B)) | equipotent(A,f153(B)). [resolve(401,b,399,b)]. 2.34/2.68 402 are_equipotent(A,f173(B)) | in(A,f173(B)) | -subset(A,f173(B)) # label(t136_zfmisc_1) # label(lemma). [clausify(332)]. 2.34/2.68 Derived: in(A,f173(B)) | -subset(A,f173(B)) | equipotent(A,f173(B)). [resolve(402,a,399,b)]. 2.34/2.68 2.34/2.68 ============================== end predicate elimination ============= 2.34/2.68 2.34/2.68 Auto_denials: (non-Horn, no changes). 2.34/2.68 2.34/2.68 Term ordering decisions: 2.34/2.68 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. ordered_pair=1. apply=1. cartesian_product2=1. relation_dom_restriction=1. relation_composition=1. set_difference=1. set_intersection2=1. relation_image=1. set_union2=1. relation_inverse_image=1. relation_restriction=1. relation_rng_restriction=1. unordered_pair=1. complements_of_subsets=1. fiber=1. subset_complement=1. meet_of_subsets=1. union_of_subsets=1. f4=1. f5=1. f6=1. f7=1. f8=1. f9=1. f12=1. f13=1. f15=1. f21=1. f22=1. f31=1. f34=1. f35=1. f36=1. f37=1. f40=1. f41=1. f56=1. f57=1. f58=1. f62=1. f64=1. f65=1. f73=1. f75=1. f76=1. f77=1. f78=1. f80=1. f81=1. f86=1. f87=1. f88=1. f93=1. f94=1. f95=1. f109=1. f110=1. f114=1. f115=1. f116=1. f117=1. f118=1. f119=1. f120=1. f121=1. f125=1. f126=1. f135=1. f139=1. f140=1. f141=1. f143=1. f150=1. f154=1. f155=1. f156=1. f165=1. f166=1. f168=1. f169=1. f170=1. f172=1. f174=1. f175=1. f180=1. f183=1. f184=1. f185=1. f186=1. singleton=1. relation_dom=1. powerset=1. relation_rng=1. relation_field=1. succ=1. union=1. identity_relation=1. function_inverse=1. inclusion_relation=1. relation_inverse=1. set_meet=1. cast_to_subset=1. pair_first=1. pair_second=1. f1=1. f23=1. f24=1. f27=1. f30=1. f46=1. f52=1. f53=1. f54=1. f55=1. f63=1. f68=1. f69=1. f70=1. f71=1. f72=1. f74=1. f83=1. f84=1. f85=1. f89=1. f90=1. f91=1. f92=1. f107=1. f108=1. f112=1. f113=1. f127=1. f130=1. f131=1. f132=1. f133=1. f134=1. f151=1. f152=1. f153=1. f157=1. f158=1. f159=1. f160=1. f161=1. f163=1. f164=1. f173=1. f181=1. f182=1. f187=1. f188=1. relation_dom_as_subset=1. relation_rng_as_subset=1. unordered_triple=1. subset_difference=1. f2=1. f3=1. f10=1. f11=1. f14=1. f18=1. f19=1. f20=1. f26=1. f28=1. f29=1. f32=1. f38=1. f39=1. f43=1. f44=1. f45=1. f47=1. f50=1. f51=1. f59=1. f60=1. f61=1. f66=1. f67=1. f79=1. f82=1. f96=1. f97=1. f98=1. f99=1. f100=1. f101=1. f102=1. f103=1. f111=1. f122=1. f123=1. f124=1. f128=1. f129=1. f136=1. f142=1. f144=1. f145=1. f146=1. f147=1. f148=1. f162=1. f167=1. f171=1. f177=1. f178=1. f179=1. f16=1. f17=1. f25=1. f33=1. f42=1. f48=1. f49=1. f104=1. f105=1. f106=1. f137=1. f138=1. f176=1. f149=1. 2.34/2.68 2.34/2.68 ============================== end of process initial clauses ======== 2.34/2.68 2.34/2.68 ============================== CLAUSES FOR SEARCH ==================== 2.34/2.68 2.34/2.68 ============================== end of clauses for search ============= 2.34/2.68 2.34/2.68 ============================== SEARCH ================================ 2.34/2.68 2.34/2.68 % Starting search at 0.73 seconds. 2.34/2.68 2.34/2.68 Low Water (keep): wt=55.000, iters=3689 2.34/2.68 2.34/2.68 Low Water (keep): wt=47.000, iters=3459 2.34/2.68 2.34/2.68 Low Water (keep): wt=46.000, iters=3465 2.34/2.68 2.34/2.68 Low Water (keep): wt=44.000, iters=3392 2.34/2.68 2.34/2.68 Low Water (keep): wt=40.000, iters=3362 2.34/2.68 2.34/2.68 Low Water (keep): wt=38.000, iters=3481 2.34/2.68 2.34/2.68 Low Water (kAlarm clock 119.71/120.08 Prover9 interrupted 119.71/120.08 EOF