%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET002-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:24:43 AM UTC 2025
% Result : Unsatisfiable 0.13s 0.39s
% Output : Proof 0.13s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET002-1 : TPTP v9.0.0. Released v1.0.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.13/0.35 % Computer : n008.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 : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Jun 20 04:00:48 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.13/0.39 % SZS status Unsatisfiable
% 0.13/0.39 % SZS output start Proof
% 0.13/0.39 tff(member_type, type, (
% 0.13/0.39 member: ( $i * $i ) > $o)).
% 0.13/0.39 tff(a_type, type, (
% 0.13/0.39 a: $i)).
% 0.13/0.39 tff(member_of_1_not_of_2_type, type, (
% 0.13/0.39 member_of_1_not_of_2: ( $i * $i ) > $i)).
% 0.13/0.39 tff(aUa_type, type, (
% 0.13/0.39 aUa: $i)).
% 0.13/0.39 tff(subset_type, type, (
% 0.13/0.39 subset: ( $i * $i ) > $o)).
% 0.13/0.39 tff(union_type, type, (
% 0.13/0.39 union: ( $i * $i * $i ) > $o)).
% 0.13/0.39 tff(equal_sets_type, type, (
% 0.13/0.39 equal_sets: ( $i * $i ) > $o)).
% 0.13/0.39 tff(1,assumption,(~subset(aUa, a)), introduced(assumption)).
% 0.13/0.39 tff(2,plain,
% 0.13/0.39 (^[Subset: $i, Superset: $i] : refl((subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset)) <=> (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset)))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(3,plain,
% 0.13/0.39 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset)) <=> ![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[2])).
% 0.13/0.39 tff(4,plain,
% 0.13/0.39 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset)) <=> ![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(5,axiom,(![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-0.ax','subsets_axiom1')).
% 0.13/0.39 tff(6,plain,
% 0.13/0.39 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.13/0.39 tff(7,plain,(
% 0.13/0.39 ![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.13/0.39 inference(skolemize,[status(sab)],[6])).
% 0.13/0.39 tff(8,plain,
% 0.13/0.39 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.13/0.39 tff(9,plain,
% 0.13/0.39 (((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(aUa, a) | member(member_of_1_not_of_2(aUa, a), aUa))) <=> ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(aUa, a) | member(member_of_1_not_of_2(aUa, a), aUa))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(10,plain,
% 0.13/0.39 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(aUa, a) | member(member_of_1_not_of_2(aUa, a), aUa))),
% 0.13/0.39 inference(quant_inst,[status(thm)],[])).
% 0.13/0.39 tff(11,plain,
% 0.13/0.39 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(aUa, a) | member(member_of_1_not_of_2(aUa, a), aUa)),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[10, 9])).
% 0.13/0.39 tff(12,plain,
% 0.13/0.39 (member(member_of_1_not_of_2(aUa, a), aUa)),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[11, 8, 1])).
% 0.13/0.39 tff(13,plain,
% 0.13/0.39 (^[Subset: $i, Superset: $i] : refl(((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset)) <=> ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset)))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(14,plain,
% 0.13/0.39 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset)) <=> ![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[13])).
% 0.13/0.39 tff(15,plain,
% 0.13/0.39 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset)) <=> ![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(16,axiom,(![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-0.ax','subsets_axiom2')).
% 0.13/0.39 tff(17,plain,
% 0.13/0.39 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.13/0.39 tff(18,plain,(
% 0.13/0.39 ![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.13/0.39 inference(skolemize,[status(sab)],[17])).
% 0.13/0.39 tff(19,plain,
% 0.13/0.39 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.13/0.39 tff(20,plain,
% 0.13/0.39 (((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(aUa, a), a)) | subset(aUa, a))) <=> ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(aUa, a), a)) | subset(aUa, a))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(21,plain,
% 0.13/0.39 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(aUa, a), a)) | subset(aUa, a))),
% 0.13/0.39 inference(quant_inst,[status(thm)],[])).
% 0.13/0.39 tff(22,plain,
% 0.13/0.39 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(aUa, a), a)) | subset(aUa, a)),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.13/0.39 tff(23,plain,
% 0.13/0.39 (~member(member_of_1_not_of_2(aUa, a), a)),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[22, 19, 1])).
% 0.13/0.39 tff(24,plain,
% 0.13/0.39 (union(a, a, aUa) <=> union(a, a, aUa)),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(25,axiom,(union(a, a, aUa)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_union_a_is_aUa')).
% 0.13/0.39 tff(26,plain,
% 0.13/0.39 (union(a, a, aUa)),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[25, 24])).
% 0.13/0.39 tff(27,plain,
% 0.13/0.39 (^[Set1: $i, Set2: $i, Union: $i, Element: $i] : refl((member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))) <=> (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(28,plain,
% 0.13/0.39 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[27])).
% 0.13/0.39 tff(29,plain,
% 0.13/0.39 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(30,plain,
% 0.13/0.39 (^[Set1: $i, Set2: $i, Union: $i, Element: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~union(Set1, Set2, Union)) | (~member(Element, Union))) <=> ((~member(Element, Union)) | (~union(Set1, Set2, Union)))), ((((~union(Set1, Set2, Union)) | (~member(Element, Union))) | member(Element, Set1)) <=> (((~member(Element, Union)) | (~union(Set1, Set2, Union))) | member(Element, Set1)))), rewrite((((~member(Element, Union)) | (~union(Set1, Set2, Union))) | member(Element, Set1)) <=> (member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))), ((((~union(Set1, Set2, Union)) | (~member(Element, Union))) | member(Element, Set1)) <=> (member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))))), (((((~union(Set1, Set2, Union)) | (~member(Element, Union))) | member(Element, Set1)) | member(Element, Set2)) <=> ((member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))) | member(Element, Set2)))), rewrite(((member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))) | member(Element, Set2)) <=> (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))), (((((~union(Set1, Set2, Union)) | (~member(Element, Union))) | member(Element, Set1)) | member(Element, Set2)) <=> (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(31,plain,
% 0.13/0.39 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((((~union(Set1, Set2, Union)) | (~member(Element, Union))) | member(Element, Set1)) | member(Element, Set2)) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[30])).
% 0.13/0.39 tff(32,axiom,(![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((((~union(Set1, Set2, Union)) | (~member(Element, Union))) | member(Element, Set1)) | member(Element, Set2))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-1.ax','member_of_union_is_member_of_one_set')).
% 0.13/0.39 tff(33,plain,
% 0.13/0.39 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.13/0.39 tff(34,plain,
% 0.13/0.39 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[33, 29])).
% 0.13/0.39 tff(35,plain,(
% 0.13/0.39 ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.13/0.39 inference(skolemize,[status(sab)],[34])).
% 0.13/0.39 tff(36,plain,
% 0.13/0.39 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[35, 28])).
% 0.13/0.39 tff(37,plain,
% 0.13/0.39 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | (member(member_of_1_not_of_2(aUa, a), a) | (~member(member_of_1_not_of_2(aUa, a), aUa)) | (~union(a, a, aUa)))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | member(member_of_1_not_of_2(aUa, a), a) | (~member(member_of_1_not_of_2(aUa, a), aUa)) | (~union(a, a, aUa)))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(38,plain,
% 0.13/0.39 ((member(member_of_1_not_of_2(aUa, a), a) | member(member_of_1_not_of_2(aUa, a), a) | (~member(member_of_1_not_of_2(aUa, a), aUa)) | (~union(a, a, aUa))) <=> (member(member_of_1_not_of_2(aUa, a), a) | (~member(member_of_1_not_of_2(aUa, a), aUa)) | (~union(a, a, aUa)))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(39,plain,
% 0.13/0.39 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | (member(member_of_1_not_of_2(aUa, a), a) | member(member_of_1_not_of_2(aUa, a), a) | (~member(member_of_1_not_of_2(aUa, a), aUa)) | (~union(a, a, aUa)))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | (member(member_of_1_not_of_2(aUa, a), a) | (~member(member_of_1_not_of_2(aUa, a), aUa)) | (~union(a, a, aUa))))),
% 0.13/0.39 inference(monotonicity,[status(thm)],[38])).
% 0.13/0.39 tff(40,plain,
% 0.13/0.39 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | (member(member_of_1_not_of_2(aUa, a), a) | member(member_of_1_not_of_2(aUa, a), a) | (~member(member_of_1_not_of_2(aUa, a), aUa)) | (~union(a, a, aUa)))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | member(member_of_1_not_of_2(aUa, a), a) | (~member(member_of_1_not_of_2(aUa, a), aUa)) | (~union(a, a, aUa)))),
% 0.13/0.40 inference(transitivity,[status(thm)],[39, 37])).
% 0.13/0.40 tff(41,plain,
% 0.13/0.40 ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | (member(member_of_1_not_of_2(aUa, a), a) | member(member_of_1_not_of_2(aUa, a), a) | (~member(member_of_1_not_of_2(aUa, a), aUa)) | (~union(a, a, aUa)))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(42,plain,
% 0.13/0.40 ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | member(member_of_1_not_of_2(aUa, a), a) | (~member(member_of_1_not_of_2(aUa, a), aUa)) | (~union(a, a, aUa))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.13/0.40 tff(43,plain,
% 0.13/0.40 ($false),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[42, 36, 26, 23, 12])).
% 0.13/0.40 tff(44,plain,(subset(aUa, a)), inference(lemma,lemma(discharge,[]))).
% 0.13/0.40 tff(45,plain,
% 0.13/0.40 ((~equal_sets(aUa, a)) <=> (~equal_sets(aUa, a))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(46,axiom,(~equal_sets(aUa, a)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_a_equals_aUa')).
% 0.13/0.40 tff(47,plain,
% 0.13/0.40 (~equal_sets(aUa, a)),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.13/0.40 tff(48,plain,
% 0.13/0.40 (^[Set1: $i, Set2: $i] : refl((equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2))) <=> (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2))))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(49,plain,
% 0.13/0.40 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2))) <=> ![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[48])).
% 0.13/0.40 tff(50,plain,
% 0.13/0.40 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2))) <=> ![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(51,plain,
% 0.13/0.40 (^[Set1: $i, Set2: $i] : trans(monotonicity(rewrite(((~subset(Set1, Set2)) | (~subset(Set2, Set1))) <=> ((~subset(Set2, Set1)) | (~subset(Set1, Set2)))), ((((~subset(Set1, Set2)) | (~subset(Set2, Set1))) | equal_sets(Set2, Set1)) <=> (((~subset(Set2, Set1)) | (~subset(Set1, Set2))) | equal_sets(Set2, Set1)))), rewrite((((~subset(Set2, Set1)) | (~subset(Set1, Set2))) | equal_sets(Set2, Set1)) <=> (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))), ((((~subset(Set1, Set2)) | (~subset(Set2, Set1))) | equal_sets(Set2, Set1)) <=> (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(52,plain,
% 0.13/0.40 (![Set1: $i, Set2: $i] : (((~subset(Set1, Set2)) | (~subset(Set2, Set1))) | equal_sets(Set2, Set1)) <=> ![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[51])).
% 0.13/0.40 tff(53,axiom,(![Set1: $i, Set2: $i] : (((~subset(Set1, Set2)) | (~subset(Set2, Set1))) | equal_sets(Set2, Set1))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-0.ax','subsets_are_set_equal_sets')).
% 0.13/0.40 tff(54,plain,
% 0.13/0.40 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.13/0.40 tff(55,plain,
% 0.13/0.40 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[54, 50])).
% 0.13/0.40 tff(56,plain,(
% 0.13/0.40 ![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.13/0.40 inference(skolemize,[status(sab)],[55])).
% 0.13/0.40 tff(57,plain,
% 0.13/0.40 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[56, 49])).
% 0.13/0.40 tff(58,plain,
% 0.13/0.40 (((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | (equal_sets(aUa, a) | (~subset(aUa, a)) | (~subset(a, aUa)))) <=> ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | equal_sets(aUa, a) | (~subset(aUa, a)) | (~subset(a, aUa)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(59,plain,
% 0.13/0.40 ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | (equal_sets(aUa, a) | (~subset(aUa, a)) | (~subset(a, aUa)))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(60,plain,
% 0.13/0.40 ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | equal_sets(aUa, a) | (~subset(aUa, a)) | (~subset(a, aUa))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.13/0.40 tff(61,plain,
% 0.13/0.40 ((~subset(aUa, a)) | (~subset(a, aUa))),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[60, 57, 47])).
% 0.13/0.40 tff(62,plain,
% 0.13/0.40 (~subset(a, aUa)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[61, 44])).
% 0.13/0.40 tff(63,plain,
% 0.13/0.40 (((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(a, aUa) | member(member_of_1_not_of_2(a, aUa), a))) <=> ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(a, aUa) | member(member_of_1_not_of_2(a, aUa), a))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(64,plain,
% 0.13/0.40 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(a, aUa) | member(member_of_1_not_of_2(a, aUa), a))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(65,plain,
% 0.13/0.40 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(a, aUa) | member(member_of_1_not_of_2(a, aUa), a)),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.13/0.40 tff(66,plain,
% 0.13/0.40 (member(member_of_1_not_of_2(a, aUa), a)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[65, 8, 62])).
% 0.13/0.40 tff(67,plain,
% 0.13/0.40 (((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(a, aUa), aUa)) | subset(a, aUa))) <=> ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(a, aUa), aUa)) | subset(a, aUa))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(68,plain,
% 0.13/0.40 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(a, aUa), aUa)) | subset(a, aUa))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(69,plain,
% 0.13/0.40 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(a, aUa), aUa)) | subset(a, aUa)),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.13/0.40 tff(70,plain,
% 0.13/0.40 (~member(member_of_1_not_of_2(a, aUa), aUa)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[69, 19, 62])).
% 0.13/0.40 tff(71,plain,
% 0.13/0.40 (^[Set1: $i, Set2: $i, Union: $i, Element: $i] : refl((member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2))) <=> (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2))))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(72,plain,
% 0.13/0.40 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2))) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[71])).
% 0.13/0.40 tff(73,plain,
% 0.13/0.40 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2))) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(74,plain,
% 0.13/0.40 (^[Set1: $i, Set2: $i, Union: $i, Element: $i] : trans(monotonicity(rewrite(((~union(Set1, Set2, Union)) | (~member(Element, Set2))) <=> ((~union(Set1, Set2, Union)) | (~member(Element, Set2)))), ((((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union)) <=> (((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union)))), rewrite((((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union)) <=> (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))), ((((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union)) <=> (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(75,plain,
% 0.13/0.40 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union)) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[74])).
% 0.13/0.40 tff(76,axiom,(![Set1: $i, Set2: $i, Union: $i, Element: $i] : (((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-1.ax','member_of_set2_is_member_of_union')).
% 0.13/0.40 tff(77,plain,
% 0.13/0.40 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.13/0.40 tff(78,plain,
% 0.13/0.40 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[77, 73])).
% 0.13/0.40 tff(79,plain,(
% 0.13/0.40 ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.13/0.40 inference(skolemize,[status(sab)],[78])).
% 0.13/0.40 tff(80,plain,
% 0.13/0.40 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[79, 72])).
% 0.13/0.40 tff(81,plain,
% 0.13/0.40 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))) | (member(member_of_1_not_of_2(a, aUa), aUa) | (~union(a, a, aUa)) | (~member(member_of_1_not_of_2(a, aUa), a)))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))) | member(member_of_1_not_of_2(a, aUa), aUa) | (~union(a, a, aUa)) | (~member(member_of_1_not_of_2(a, aUa), a)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(82,plain,
% 0.13/0.40 ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))) | (member(member_of_1_not_of_2(a, aUa), aUa) | (~union(a, a, aUa)) | (~member(member_of_1_not_of_2(a, aUa), a)))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(83,plain,
% 0.13/0.40 ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))) | member(member_of_1_not_of_2(a, aUa), aUa) | (~union(a, a, aUa)) | (~member(member_of_1_not_of_2(a, aUa), a))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.13/0.40 tff(84,plain,
% 0.13/0.40 ($false),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[83, 80, 26, 70, 66])).
% 0.13/0.40 % SZS output end Proof
% 0.13/0.40 % E exiting
%------------------------------------------------------------------------------