TSTP Solution File: SET012-1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET012-1 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n003.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 : Tue Sep 20 05:04:42 EDT 2022
% Result : Unsatisfiable 0.12s 0.39s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : SET012-1 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.06/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n003.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Sat Sep 3 01:02:55 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.35 Usage: tptp [options] [-file:]file
% 0.12/0.35 -h, -? prints this message.
% 0.12/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.35 -m, -model generate model.
% 0.12/0.35 -p, -proof generate proof.
% 0.12/0.35 -c, -core generate unsat core of named formulas.
% 0.12/0.35 -st, -statistics display statistics.
% 0.12/0.35 -t:timeout set timeout (in second).
% 0.12/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.35 -<param>:<value> configuration parameter and value.
% 0.12/0.35 -o:<output-file> file to place output in.
% 0.12/0.39 % SZS status Unsatisfiable
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(member_type, type, (
% 0.12/0.39 member: ( $i * $i ) > $o)).
% 0.12/0.39 tff(complement_type, type, (
% 0.12/0.39 complement: $i > $i)).
% 0.12/0.39 tff(a_type, type, (
% 0.12/0.39 a: $i)).
% 0.12/0.39 tff(member_of_1_not_of_2_type, type, (
% 0.12/0.39 member_of_1_not_of_2: ( $i * $i ) > $i)).
% 0.12/0.39 tff(subset_type, type, (
% 0.12/0.39 subset: ( $i * $i ) > $o)).
% 0.12/0.39 tff(equal_sets_type, type, (
% 0.12/0.39 equal_sets: ( $i * $i ) > $o)).
% 0.12/0.39 tff(1,assumption,(~subset(a, complement(complement(a)))), introduced(assumption)).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 (^[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)))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (![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))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[2])).
% 0.12/0.39 tff(4,plain,
% 0.12/0.39 (![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))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(5,axiom,(![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))), file('/export/starexec/sandbox2/benchmark/Axioms/SET002-0.ax','subsets_axiom2')).
% 0.12/0.39 tff(6,plain,
% 0.12/0.39 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.12/0.39 tff(7,plain,(
% 0.12/0.39 ![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.12/0.39 inference(skolemize,[status(sab)],[6])).
% 0.12/0.39 tff(8,plain,
% 0.12/0.39 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.12/0.39 tff(9,plain,
% 0.12/0.39 (((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(a, complement(complement(a))), complement(complement(a)))) | subset(a, complement(complement(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(a, complement(complement(a))), complement(complement(a)))) | subset(a, complement(complement(a))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(10,plain,
% 0.12/0.39 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(a, complement(complement(a))), complement(complement(a)))) | subset(a, complement(complement(a))))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(11,plain,
% 0.12/0.39 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(a, complement(complement(a))), complement(complement(a)))) | subset(a, complement(complement(a)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[10, 9])).
% 0.12/0.39 tff(12,plain,
% 0.12/0.39 (~member(member_of_1_not_of_2(a, complement(complement(a))), complement(complement(a)))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[11, 8, 1])).
% 0.12/0.39 tff(13,plain,
% 0.12/0.39 (^[Xs: $i, X: $i] : refl((member(X, Xs) | member(X, complement(Xs))) <=> (member(X, Xs) | member(X, complement(Xs))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(14,plain,
% 0.12/0.39 (![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs))) <=> ![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[13])).
% 0.12/0.39 tff(15,plain,
% 0.12/0.39 (![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs))) <=> ![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(16,axiom,(![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET002-0.ax','member_of_set_or_complement')).
% 0.20/0.40 tff(17,plain,
% 0.20/0.40 (![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.20/0.40 tff(18,plain,(
% 0.20/0.40 ![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.20/0.40 inference(skolemize,[status(sab)],[17])).
% 0.20/0.40 tff(19,plain,
% 0.20/0.40 (![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.20/0.40 tff(20,plain,
% 0.20/0.40 (((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | (member(member_of_1_not_of_2(a, complement(complement(a))), complement(a)) | member(member_of_1_not_of_2(a, complement(complement(a))), complement(complement(a))))) <=> ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | member(member_of_1_not_of_2(a, complement(complement(a))), complement(a)) | member(member_of_1_not_of_2(a, complement(complement(a))), complement(complement(a))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(21,plain,
% 0.20/0.40 ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | (member(member_of_1_not_of_2(a, complement(complement(a))), complement(a)) | member(member_of_1_not_of_2(a, complement(complement(a))), complement(complement(a))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(22,plain,
% 0.20/0.40 ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | member(member_of_1_not_of_2(a, complement(complement(a))), complement(a)) | member(member_of_1_not_of_2(a, complement(complement(a))), complement(complement(a)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.20/0.40 tff(23,plain,
% 0.20/0.40 (member(member_of_1_not_of_2(a, complement(complement(a))), complement(a))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[22, 19, 12])).
% 0.20/0.40 tff(24,plain,
% 0.20/0.40 (^[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)))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(25,plain,
% 0.20/0.40 (![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))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[24])).
% 0.20/0.40 tff(26,plain,
% 0.20/0.40 (![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))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(27,axiom,(![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))), file('/export/starexec/sandbox2/benchmark/Axioms/SET002-0.ax','subsets_axiom1')).
% 0.20/0.40 tff(28,plain,
% 0.20/0.40 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.20/0.40 tff(29,plain,(
% 0.20/0.40 ![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.20/0.40 inference(skolemize,[status(sab)],[28])).
% 0.20/0.40 tff(30,plain,
% 0.20/0.40 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[29, 25])).
% 0.20/0.40 tff(31,plain,
% 0.20/0.40 (((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(a, complement(complement(a))) | member(member_of_1_not_of_2(a, complement(complement(a))), a))) <=> ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(a, complement(complement(a))) | member(member_of_1_not_of_2(a, complement(complement(a))), a))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(32,plain,
% 0.20/0.40 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(a, complement(complement(a))) | member(member_of_1_not_of_2(a, complement(complement(a))), a))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(33,plain,
% 0.20/0.40 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(a, complement(complement(a))) | member(member_of_1_not_of_2(a, complement(complement(a))), a)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.20/0.40 tff(34,plain,
% 0.20/0.40 (member(member_of_1_not_of_2(a, complement(complement(a))), a)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[33, 30, 1])).
% 0.20/0.40 tff(35,plain,
% 0.20/0.40 (^[Xs: $i, X: $i] : refl(((~member(X, Xs)) | (~member(X, complement(Xs)))) <=> ((~member(X, Xs)) | (~member(X, complement(Xs)))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(36,plain,
% 0.20/0.40 (![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs)))) <=> ![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[35])).
% 0.20/0.40 tff(37,plain,
% 0.20/0.40 (![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs)))) <=> ![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(38,axiom,(![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET002-0.ax','not_member_of_set_and_complement')).
% 0.20/0.40 tff(39,plain,
% 0.20/0.40 (![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.20/0.40 tff(40,plain,(
% 0.20/0.40 ![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[39])).
% 0.20/0.40 tff(41,plain,
% 0.20/0.40 (![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[40, 36])).
% 0.20/0.40 tff(42,plain,
% 0.20/0.40 (((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | ((~member(member_of_1_not_of_2(a, complement(complement(a))), a)) | (~member(member_of_1_not_of_2(a, complement(complement(a))), complement(a))))) <=> ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | (~member(member_of_1_not_of_2(a, complement(complement(a))), a)) | (~member(member_of_1_not_of_2(a, complement(complement(a))), complement(a))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(43,plain,
% 0.20/0.40 ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | ((~member(member_of_1_not_of_2(a, complement(complement(a))), a)) | (~member(member_of_1_not_of_2(a, complement(complement(a))), complement(a))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(44,plain,
% 0.20/0.40 ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | (~member(member_of_1_not_of_2(a, complement(complement(a))), a)) | (~member(member_of_1_not_of_2(a, complement(complement(a))), complement(a)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.20/0.40 tff(45,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[44, 41, 34, 23])).
% 0.20/0.40 tff(46,plain,(subset(a, complement(complement(a)))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.40 tff(47,plain,
% 0.20/0.40 ((~equal_sets(complement(complement(a)), a)) <=> (~equal_sets(complement(complement(a)), a))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(48,axiom,(~equal_sets(complement(complement(a)), a)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_involution')).
% 0.20/0.40 tff(49,plain,
% 0.20/0.40 (~equal_sets(complement(complement(a)), a)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[48, 47])).
% 0.20/0.40 tff(50,plain,
% 0.20/0.40 (^[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))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(51,plain,
% 0.20/0.40 (![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)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[50])).
% 0.20/0.40 tff(52,plain,
% 0.20/0.40 (![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)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(53,plain,
% 0.20/0.40 (^[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)))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(54,plain,
% 0.20/0.40 (![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)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[53])).
% 0.20/0.40 tff(55,axiom,(![Set1: $i, Set2: $i] : (((~subset(Set1, Set2)) | (~subset(Set2, Set1))) | equal_sets(Set2, Set1))), file('/export/starexec/sandbox2/benchmark/Axioms/SET002-0.ax','subsets_are_set_equal_sets')).
% 0.20/0.40 tff(56,plain,
% 0.20/0.40 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[55, 54])).
% 0.20/0.40 tff(57,plain,
% 0.20/0.40 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[56, 52])).
% 0.20/0.40 tff(58,plain,(
% 0.20/0.40 ![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.40 inference(skolemize,[status(sab)],[57])).
% 0.20/0.40 tff(59,plain,
% 0.20/0.40 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[58, 51])).
% 0.20/0.40 tff(60,plain,
% 0.20/0.40 (((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | (equal_sets(complement(complement(a)), a) | (~subset(complement(complement(a)), a)) | (~subset(a, complement(complement(a)))))) <=> ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | equal_sets(complement(complement(a)), a) | (~subset(complement(complement(a)), a)) | (~subset(a, complement(complement(a)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(61,plain,
% 0.20/0.40 ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | (equal_sets(complement(complement(a)), a) | (~subset(complement(complement(a)), a)) | (~subset(a, complement(complement(a)))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(62,plain,
% 0.20/0.40 ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | equal_sets(complement(complement(a)), a) | (~subset(complement(complement(a)), a)) | (~subset(a, complement(complement(a))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.20/0.40 tff(63,plain,
% 0.20/0.40 ((~subset(complement(complement(a)), a)) | (~subset(a, complement(complement(a))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[62, 59, 49])).
% 0.20/0.40 tff(64,plain,
% 0.20/0.40 (~subset(complement(complement(a)), a)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[63, 46])).
% 0.20/0.40 tff(65,plain,
% 0.20/0.40 (((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(complement(complement(a)), a) | member(member_of_1_not_of_2(complement(complement(a)), a), complement(complement(a))))) <=> ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(complement(complement(a)), a) | member(member_of_1_not_of_2(complement(complement(a)), a), complement(complement(a))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(66,plain,
% 0.20/0.40 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(complement(complement(a)), a) | member(member_of_1_not_of_2(complement(complement(a)), a), complement(complement(a))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(67,plain,
% 0.20/0.40 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(complement(complement(a)), a) | member(member_of_1_not_of_2(complement(complement(a)), a), complement(complement(a)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.20/0.40 tff(68,plain,
% 0.20/0.40 (member(member_of_1_not_of_2(complement(complement(a)), a), complement(complement(a)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[67, 30, 64])).
% 0.20/0.40 tff(69,plain,
% 0.20/0.40 (((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | ((~member(member_of_1_not_of_2(complement(complement(a)), a), complement(a))) | (~member(member_of_1_not_of_2(complement(complement(a)), a), complement(complement(a)))))) <=> ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | (~member(member_of_1_not_of_2(complement(complement(a)), a), complement(a))) | (~member(member_of_1_not_of_2(complement(complement(a)), a), complement(complement(a)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(70,plain,
% 0.20/0.40 ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | ((~member(member_of_1_not_of_2(complement(complement(a)), a), complement(a))) | (~member(member_of_1_not_of_2(complement(complement(a)), a), complement(complement(a)))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(71,plain,
% 0.20/0.40 ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | (~member(member_of_1_not_of_2(complement(complement(a)), a), complement(a))) | (~member(member_of_1_not_of_2(complement(complement(a)), a), complement(complement(a))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.20/0.40 tff(72,plain,
% 0.20/0.40 (~member(member_of_1_not_of_2(complement(complement(a)), a), complement(a))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[71, 41, 68])).
% 0.20/0.40 tff(73,plain,
% 0.20/0.40 (((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(complement(complement(a)), a), a)) | subset(complement(complement(a)), 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(complement(complement(a)), a), a)) | subset(complement(complement(a)), a))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(74,plain,
% 0.20/0.40 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(complement(complement(a)), a), a)) | subset(complement(complement(a)), a))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(75,plain,
% 0.20/0.40 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(complement(complement(a)), a), a)) | subset(complement(complement(a)), a)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.20/0.40 tff(76,plain,
% 0.20/0.40 (~member(member_of_1_not_of_2(complement(complement(a)), a), a)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[75, 8, 64])).
% 0.20/0.40 tff(77,plain,
% 0.20/0.40 (((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | (member(member_of_1_not_of_2(complement(complement(a)), a), a) | member(member_of_1_not_of_2(complement(complement(a)), a), complement(a)))) <=> ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | member(member_of_1_not_of_2(complement(complement(a)), a), a) | member(member_of_1_not_of_2(complement(complement(a)), a), complement(a)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(78,plain,
% 0.20/0.40 ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | (member(member_of_1_not_of_2(complement(complement(a)), a), a) | member(member_of_1_not_of_2(complement(complement(a)), a), complement(a)))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(79,plain,
% 0.20/0.41 ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | member(member_of_1_not_of_2(complement(complement(a)), a), a) | member(member_of_1_not_of_2(complement(complement(a)), a), complement(a))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.20/0.41 tff(80,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[79, 19, 76, 72])).
% 0.20/0.41 % SZS output end Proof
%------------------------------------------------------------------------------