TSTP Solution File: SET012-2 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET012-2 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n014.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 05:04:43 EDT 2022
% Result : Unsatisfiable 0.21s 0.46s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET012-2 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.34 % Computer : n014.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Sat Sep 3 01:03:31 EDT 2022
% 0.14/0.34 % CPUTime :
% 0.14/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.14/0.35 Usage: tptp [options] [-file:]file
% 0.14/0.35 -h, -? prints this message.
% 0.14/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.14/0.35 -m, -model generate model.
% 0.14/0.35 -p, -proof generate proof.
% 0.14/0.35 -c, -core generate unsat core of named formulas.
% 0.14/0.35 -st, -statistics display statistics.
% 0.14/0.35 -t:timeout set timeout (in second).
% 0.14/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.14/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.14/0.35 -<param>:<value> configuration parameter and value.
% 0.14/0.35 -o:<output-file> file to place output in.
% 0.21/0.46 % SZS status Unsatisfiable
% 0.21/0.46 % SZS output start Proof
% 0.21/0.46 tff(member_type, type, (
% 0.21/0.46 member: ( $i * $i ) > $o)).
% 0.21/0.46 tff(b_type, type, (
% 0.21/0.46 b: $i)).
% 0.21/0.46 tff(member_of_1_not_of_2_type, type, (
% 0.21/0.46 member_of_1_not_of_2: ( $i * $i ) > $i)).
% 0.21/0.46 tff(a_type, type, (
% 0.21/0.46 a: $i)).
% 0.21/0.46 tff(c_type, type, (
% 0.21/0.46 c: $i)).
% 0.21/0.46 tff(complement_type, type, (
% 0.21/0.46 complement: $i > $i)).
% 0.21/0.46 tff(subset_type, type, (
% 0.21/0.46 subset: ( $i * $i ) > $o)).
% 0.21/0.46 tff(equal_sets_type, type, (
% 0.21/0.46 equal_sets: ( $i * $i ) > $o)).
% 0.21/0.46 tff(1,assumption,(~subset(a, c)), introduced(assumption)).
% 0.21/0.46 tff(2,plain,
% 0.21/0.46 (^[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)))),
% 0.21/0.46 inference(bind,[status(th)],[])).
% 0.21/0.46 tff(3,plain,
% 0.21/0.46 (![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))),
% 0.21/0.46 inference(quant_intro,[status(thm)],[2])).
% 0.21/0.46 tff(4,plain,
% 0.21/0.46 (![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))),
% 0.21/0.46 inference(rewrite,[status(thm)],[])).
% 0.21/0.46 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/SET002-0.ax','subsets_axiom1')).
% 0.21/0.46 tff(6,plain,
% 0.21/0.46 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.21/0.46 tff(7,plain,(
% 0.21/0.46 ![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.21/0.46 inference(skolemize,[status(sab)],[6])).
% 0.21/0.46 tff(8,plain,
% 0.21/0.46 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.21/0.46 tff(9,plain,
% 0.21/0.46 (((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(a, c) | member(member_of_1_not_of_2(a, c), a))) <=> ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(a, c) | member(member_of_1_not_of_2(a, c), a))),
% 0.21/0.46 inference(rewrite,[status(thm)],[])).
% 0.21/0.46 tff(10,plain,
% 0.21/0.46 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(a, c) | member(member_of_1_not_of_2(a, c), a))),
% 0.21/0.46 inference(quant_inst,[status(thm)],[])).
% 0.21/0.46 tff(11,plain,
% 0.21/0.46 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(a, c) | member(member_of_1_not_of_2(a, c), a)),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[10, 9])).
% 0.21/0.46 tff(12,plain,
% 0.21/0.46 (subset(a, c) | member(member_of_1_not_of_2(a, c), a)),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[11, 8])).
% 0.21/0.46 tff(13,plain,
% 0.21/0.46 (member(member_of_1_not_of_2(a, c), a)),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[12, 1])).
% 0.21/0.46 tff(14,plain,
% 0.21/0.46 (^[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)))),
% 0.21/0.46 inference(bind,[status(th)],[])).
% 0.21/0.46 tff(15,plain,
% 0.21/0.46 (![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))),
% 0.21/0.46 inference(quant_intro,[status(thm)],[14])).
% 0.21/0.46 tff(16,plain,
% 0.21/0.46 (![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))),
% 0.21/0.46 inference(rewrite,[status(thm)],[])).
% 0.21/0.46 tff(17,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')).
% 0.21/0.46 tff(18,plain,
% 0.21/0.46 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.21/0.46 tff(19,plain,(
% 0.21/0.46 ![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.21/0.46 inference(skolemize,[status(sab)],[18])).
% 0.21/0.46 tff(20,plain,
% 0.21/0.46 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[19, 15])).
% 0.21/0.46 tff(21,plain,
% 0.21/0.46 (((~![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, c), c)) | subset(a, c))) <=> ((~![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, c), c)) | subset(a, c))),
% 0.21/0.46 inference(rewrite,[status(thm)],[])).
% 0.21/0.46 tff(22,plain,
% 0.21/0.46 ((~![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, c), c)) | subset(a, c))),
% 0.21/0.46 inference(quant_inst,[status(thm)],[])).
% 0.21/0.46 tff(23,plain,
% 0.21/0.46 ((~![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, c), c)) | subset(a, c)),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.21/0.46 tff(24,plain,
% 0.21/0.46 ((~member(member_of_1_not_of_2(a, c), c)) | subset(a, c)),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[23, 20])).
% 0.21/0.46 tff(25,plain,
% 0.21/0.46 (~member(member_of_1_not_of_2(a, c), c)),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[24, 1])).
% 0.21/0.46 tff(26,plain,
% 0.21/0.46 (equal_sets(complement(b), c) <=> equal_sets(complement(b), c)),
% 0.21/0.46 inference(rewrite,[status(thm)],[])).
% 0.21/0.46 tff(27,axiom,(equal_sets(complement(b), c)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','complement_of_b_is_c')).
% 0.21/0.46 tff(28,plain,
% 0.21/0.46 (equal_sets(complement(b), c)),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.21/0.46 tff(29,plain,
% 0.21/0.46 (^[Subset: $i, Superset: $i] : refl(((~equal_sets(Subset, Superset)) | subset(Subset, Superset)) <=> ((~equal_sets(Subset, Superset)) | subset(Subset, Superset)))),
% 0.21/0.46 inference(bind,[status(th)],[])).
% 0.21/0.46 tff(30,plain,
% 0.21/0.46 (![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset)) <=> ![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))),
% 0.21/0.46 inference(quant_intro,[status(thm)],[29])).
% 0.21/0.46 tff(31,plain,
% 0.21/0.46 (![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset)) <=> ![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))),
% 0.21/0.46 inference(rewrite,[status(thm)],[])).
% 0.21/0.46 tff(32,axiom,(![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))), file('/export/starexec/sandbox2/benchmark/Axioms/SET002-0.ax','set_equal_sets_are_subsets1')).
% 0.21/0.46 tff(33,plain,
% 0.21/0.46 (![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.21/0.46 tff(34,plain,(
% 0.21/0.46 ![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))),
% 0.21/0.46 inference(skolemize,[status(sab)],[33])).
% 0.21/0.46 tff(35,plain,
% 0.21/0.46 (![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[34, 30])).
% 0.21/0.46 tff(36,plain,
% 0.21/0.46 (((~![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))) | ((~equal_sets(complement(b), c)) | subset(complement(b), c))) <=> ((~![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))) | (~equal_sets(complement(b), c)) | subset(complement(b), c))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(37,plain,
% 0.21/0.47 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))) | ((~equal_sets(complement(b), c)) | subset(complement(b), c))),
% 0.21/0.47 inference(quant_inst,[status(thm)],[])).
% 0.21/0.47 tff(38,plain,
% 0.21/0.47 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))) | (~equal_sets(complement(b), c)) | subset(complement(b), c)),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.21/0.47 tff(39,plain,
% 0.21/0.47 (subset(complement(b), c)),
% 0.21/0.47 inference(unit_resolution,[status(thm)],[38, 35, 28])).
% 0.21/0.47 tff(40,plain,
% 0.21/0.47 (^[Subset: $i, Element: $i, Superset: $i] : refl((member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset))) <=> (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset))))),
% 0.21/0.47 inference(bind,[status(th)],[])).
% 0.21/0.47 tff(41,plain,
% 0.21/0.47 (![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset))) <=> ![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))),
% 0.21/0.47 inference(quant_intro,[status(thm)],[40])).
% 0.21/0.47 tff(42,plain,
% 0.21/0.47 (![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset))) <=> ![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(43,plain,
% 0.21/0.47 (^[Subset: $i, Element: $i, Superset: $i] : rewrite((((~member(Element, Subset)) | (~subset(Subset, Superset))) | member(Element, Superset)) <=> (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset))))),
% 0.21/0.47 inference(bind,[status(th)],[])).
% 0.21/0.47 tff(44,plain,
% 0.21/0.47 (![Subset: $i, Element: $i, Superset: $i] : (((~member(Element, Subset)) | (~subset(Subset, Superset))) | member(Element, Superset)) <=> ![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))),
% 0.21/0.47 inference(quant_intro,[status(thm)],[43])).
% 0.21/0.47 tff(45,axiom,(![Subset: $i, Element: $i, Superset: $i] : (((~member(Element, Subset)) | (~subset(Subset, Superset))) | member(Element, Superset))), file('/export/starexec/sandbox2/benchmark/Axioms/SET002-0.ax','membership_in_subsets')).
% 0.21/0.47 tff(46,plain,
% 0.21/0.47 (![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.21/0.47 tff(47,plain,
% 0.21/0.47 (![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[46, 42])).
% 0.21/0.47 tff(48,plain,(
% 0.21/0.47 ![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))),
% 0.21/0.47 inference(skolemize,[status(sab)],[47])).
% 0.21/0.47 tff(49,plain,
% 0.21/0.47 (![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[48, 41])).
% 0.21/0.47 tff(50,plain,
% 0.21/0.47 (((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(a, c), c) | (~subset(complement(b), c)) | (~member(member_of_1_not_of_2(a, c), complement(b))))) <=> ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | member(member_of_1_not_of_2(a, c), c) | (~subset(complement(b), c)) | (~member(member_of_1_not_of_2(a, c), complement(b))))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(51,plain,
% 0.21/0.47 ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(a, c), c) | (~subset(complement(b), c)) | (~member(member_of_1_not_of_2(a, c), complement(b))))),
% 0.21/0.47 inference(quant_inst,[status(thm)],[])).
% 0.21/0.47 tff(52,plain,
% 0.21/0.47 ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | member(member_of_1_not_of_2(a, c), c) | (~subset(complement(b), c)) | (~member(member_of_1_not_of_2(a, c), complement(b)))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.21/0.47 tff(53,plain,
% 0.21/0.47 (~member(member_of_1_not_of_2(a, c), complement(b))),
% 0.21/0.47 inference(unit_resolution,[status(thm)],[52, 49, 39, 25])).
% 0.21/0.47 tff(54,plain,
% 0.21/0.47 (^[Xs: $i, X: $i] : refl((member(X, Xs) | member(X, complement(Xs))) <=> (member(X, Xs) | member(X, complement(Xs))))),
% 0.21/0.47 inference(bind,[status(th)],[])).
% 0.21/0.47 tff(55,plain,
% 0.21/0.47 (![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs))) <=> ![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.21/0.47 inference(quant_intro,[status(thm)],[54])).
% 0.21/0.47 tff(56,plain,
% 0.21/0.47 (![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs))) <=> ![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(57,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')).
% 0.21/0.47 tff(58,plain,
% 0.21/0.47 (![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.21/0.47 tff(59,plain,(
% 0.21/0.47 ![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.21/0.47 inference(skolemize,[status(sab)],[58])).
% 0.21/0.47 tff(60,plain,
% 0.21/0.47 (![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[59, 55])).
% 0.21/0.47 tff(61,plain,
% 0.21/0.47 (((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | (member(member_of_1_not_of_2(a, c), b) | member(member_of_1_not_of_2(a, c), complement(b)))) <=> ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | member(member_of_1_not_of_2(a, c), b) | member(member_of_1_not_of_2(a, c), complement(b)))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(62,plain,
% 0.21/0.47 ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | (member(member_of_1_not_of_2(a, c), b) | member(member_of_1_not_of_2(a, c), complement(b)))),
% 0.21/0.47 inference(quant_inst,[status(thm)],[])).
% 0.21/0.47 tff(63,plain,
% 0.21/0.47 ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | member(member_of_1_not_of_2(a, c), b) | member(member_of_1_not_of_2(a, c), complement(b))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.21/0.47 tff(64,plain,
% 0.21/0.47 (member(member_of_1_not_of_2(a, c), b)),
% 0.21/0.47 inference(unit_resolution,[status(thm)],[63, 60, 53])).
% 0.21/0.47 tff(65,plain,
% 0.21/0.47 (equal_sets(complement(a), b) <=> equal_sets(complement(a), b)),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(66,axiom,(equal_sets(complement(a), b)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','complement_of_a_is_b')).
% 0.21/0.47 tff(67,plain,
% 0.21/0.47 (equal_sets(complement(a), b)),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.21/0.47 tff(68,plain,
% 0.21/0.47 (^[Subset: $i, Superset: $i] : refl(((~equal_sets(Superset, Subset)) | subset(Subset, Superset)) <=> ((~equal_sets(Superset, Subset)) | subset(Subset, Superset)))),
% 0.21/0.47 inference(bind,[status(th)],[])).
% 0.21/0.47 tff(69,plain,
% 0.21/0.47 (![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset)) <=> ![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))),
% 0.21/0.47 inference(quant_intro,[status(thm)],[68])).
% 0.21/0.47 tff(70,plain,
% 0.21/0.47 (![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset)) <=> ![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(71,axiom,(![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))), file('/export/starexec/sandbox2/benchmark/Axioms/SET002-0.ax','set_equal_sets_are_subsets2')).
% 0.21/0.47 tff(72,plain,
% 0.21/0.47 (![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.21/0.47 tff(73,plain,(
% 0.21/0.47 ![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))),
% 0.21/0.47 inference(skolemize,[status(sab)],[72])).
% 0.21/0.47 tff(74,plain,
% 0.21/0.47 (![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[73, 69])).
% 0.21/0.47 tff(75,plain,
% 0.21/0.47 (((~![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))) | ((~equal_sets(complement(a), b)) | subset(b, complement(a)))) <=> ((~![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))) | (~equal_sets(complement(a), b)) | subset(b, complement(a)))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(76,plain,
% 0.21/0.47 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))) | ((~equal_sets(complement(a), b)) | subset(b, complement(a)))),
% 0.21/0.47 inference(quant_inst,[status(thm)],[])).
% 0.21/0.47 tff(77,plain,
% 0.21/0.47 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))) | (~equal_sets(complement(a), b)) | subset(b, complement(a))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.21/0.47 tff(78,plain,
% 0.21/0.47 (subset(b, complement(a))),
% 0.21/0.47 inference(unit_resolution,[status(thm)],[77, 74, 67])).
% 0.21/0.47 tff(79,plain,
% 0.21/0.47 (((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | ((~subset(b, complement(a))) | member(member_of_1_not_of_2(a, c), complement(a)) | (~member(member_of_1_not_of_2(a, c), b)))) <=> ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (~subset(b, complement(a))) | member(member_of_1_not_of_2(a, c), complement(a)) | (~member(member_of_1_not_of_2(a, c), b)))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(80,plain,
% 0.21/0.47 ((member(member_of_1_not_of_2(a, c), complement(a)) | (~subset(b, complement(a))) | (~member(member_of_1_not_of_2(a, c), b))) <=> ((~subset(b, complement(a))) | member(member_of_1_not_of_2(a, c), complement(a)) | (~member(member_of_1_not_of_2(a, c), b)))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(81,plain,
% 0.21/0.47 (((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(a, c), complement(a)) | (~subset(b, complement(a))) | (~member(member_of_1_not_of_2(a, c), b)))) <=> ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | ((~subset(b, complement(a))) | member(member_of_1_not_of_2(a, c), complement(a)) | (~member(member_of_1_not_of_2(a, c), b))))),
% 0.21/0.47 inference(monotonicity,[status(thm)],[80])).
% 0.21/0.47 tff(82,plain,
% 0.21/0.47 (((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(a, c), complement(a)) | (~subset(b, complement(a))) | (~member(member_of_1_not_of_2(a, c), b)))) <=> ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (~subset(b, complement(a))) | member(member_of_1_not_of_2(a, c), complement(a)) | (~member(member_of_1_not_of_2(a, c), b)))),
% 0.21/0.47 inference(transitivity,[status(thm)],[81, 79])).
% 0.21/0.47 tff(83,plain,
% 0.21/0.47 ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(a, c), complement(a)) | (~subset(b, complement(a))) | (~member(member_of_1_not_of_2(a, c), b)))),
% 0.21/0.47 inference(quant_inst,[status(thm)],[])).
% 0.21/0.47 tff(84,plain,
% 0.21/0.47 ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (~subset(b, complement(a))) | member(member_of_1_not_of_2(a, c), complement(a)) | (~member(member_of_1_not_of_2(a, c), b))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.21/0.47 tff(85,plain,
% 0.21/0.47 (member(member_of_1_not_of_2(a, c), complement(a)) | (~member(member_of_1_not_of_2(a, c), b))),
% 0.21/0.47 inference(unit_resolution,[status(thm)],[84, 49, 78])).
% 0.21/0.47 tff(86,plain,
% 0.21/0.47 (member(member_of_1_not_of_2(a, c), complement(a))),
% 0.21/0.47 inference(unit_resolution,[status(thm)],[85, 64])).
% 0.21/0.47 tff(87,plain,
% 0.21/0.47 (^[Xs: $i, X: $i] : refl(((~member(X, Xs)) | (~member(X, complement(Xs)))) <=> ((~member(X, Xs)) | (~member(X, complement(Xs)))))),
% 0.21/0.47 inference(bind,[status(th)],[])).
% 0.21/0.47 tff(88,plain,
% 0.21/0.47 (![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs)))) <=> ![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.21/0.47 inference(quant_intro,[status(thm)],[87])).
% 0.21/0.47 tff(89,plain,
% 0.21/0.47 (![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs)))) <=> ![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(90,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')).
% 0.21/0.47 tff(91,plain,
% 0.21/0.47 (![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.21/0.47 tff(92,plain,(
% 0.21/0.47 ![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.21/0.47 inference(skolemize,[status(sab)],[91])).
% 0.21/0.47 tff(93,plain,
% 0.21/0.47 (![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[92, 88])).
% 0.21/0.47 tff(94,plain,
% 0.21/0.47 (((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | ((~member(member_of_1_not_of_2(a, c), a)) | (~member(member_of_1_not_of_2(a, c), complement(a))))) <=> ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | (~member(member_of_1_not_of_2(a, c), a)) | (~member(member_of_1_not_of_2(a, c), complement(a))))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(95,plain,
% 0.21/0.47 ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | ((~member(member_of_1_not_of_2(a, c), a)) | (~member(member_of_1_not_of_2(a, c), complement(a))))),
% 0.21/0.47 inference(quant_inst,[status(thm)],[])).
% 0.21/0.47 tff(96,plain,
% 0.21/0.47 ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | (~member(member_of_1_not_of_2(a, c), a)) | (~member(member_of_1_not_of_2(a, c), complement(a)))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.21/0.47 tff(97,plain,
% 0.21/0.47 ($false),
% 0.21/0.47 inference(unit_resolution,[status(thm)],[96, 93, 86, 13])).
% 0.21/0.47 tff(98,plain,(subset(a, c)), inference(lemma,lemma(discharge,[]))).
% 0.21/0.47 tff(99,plain,
% 0.21/0.47 ((~equal_sets(a, c)) <=> (~equal_sets(a, c))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(100,axiom,(~equal_sets(a, c)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_a_equals_c')).
% 0.21/0.47 tff(101,plain,
% 0.21/0.47 (~equal_sets(a, c)),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.21/0.47 tff(102,plain,
% 0.21/0.47 (^[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))))),
% 0.21/0.47 inference(bind,[status(th)],[])).
% 0.21/0.47 tff(103,plain,
% 0.21/0.47 (![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)))),
% 0.21/0.47 inference(quant_intro,[status(thm)],[102])).
% 0.21/0.47 tff(104,plain,
% 0.21/0.47 (![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)))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(105,plain,
% 0.21/0.47 (^[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)))))),
% 0.21/0.47 inference(bind,[status(th)],[])).
% 0.21/0.47 tff(106,plain,
% 0.21/0.47 (![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)))),
% 0.21/0.47 inference(quant_intro,[status(thm)],[105])).
% 0.21/0.47 tff(107,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')).
% 0.21/0.47 tff(108,plain,
% 0.21/0.47 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.21/0.47 tff(109,plain,
% 0.21/0.47 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[108, 104])).
% 0.21/0.47 tff(110,plain,(
% 0.21/0.47 ![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.21/0.47 inference(skolemize,[status(sab)],[109])).
% 0.21/0.47 tff(111,plain,
% 0.21/0.47 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[110, 103])).
% 0.21/0.47 tff(112,plain,
% 0.21/0.47 (((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | (equal_sets(a, c) | (~subset(a, c)) | (~subset(c, a)))) <=> ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | equal_sets(a, c) | (~subset(a, c)) | (~subset(c, a)))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(113,plain,
% 0.21/0.47 ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | (equal_sets(a, c) | (~subset(a, c)) | (~subset(c, a)))),
% 0.21/0.47 inference(quant_inst,[status(thm)],[])).
% 0.21/0.47 tff(114,plain,
% 0.21/0.47 ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | equal_sets(a, c) | (~subset(a, c)) | (~subset(c, a))),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[113, 112])).
% 0.21/0.47 tff(115,plain,
% 0.21/0.47 ((~subset(a, c)) | (~subset(c, a))),
% 0.21/0.47 inference(unit_resolution,[status(thm)],[114, 111, 101])).
% 0.21/0.47 tff(116,plain,
% 0.21/0.47 (~subset(c, a)),
% 0.21/0.47 inference(unit_resolution,[status(thm)],[115, 98])).
% 0.21/0.47 tff(117,plain,
% 0.21/0.47 (((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(c, a) | member(member_of_1_not_of_2(c, a), c))) <=> ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(c, a) | member(member_of_1_not_of_2(c, a), c))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(118,plain,
% 0.21/0.47 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(c, a) | member(member_of_1_not_of_2(c, a), c))),
% 0.21/0.47 inference(quant_inst,[status(thm)],[])).
% 0.21/0.47 tff(119,plain,
% 0.21/0.47 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(c, a) | member(member_of_1_not_of_2(c, a), c)),
% 0.21/0.47 inference(modus_ponens,[status(thm)],[118, 117])).
% 0.21/0.47 tff(120,plain,
% 0.21/0.47 (subset(c, a) | member(member_of_1_not_of_2(c, a), c)),
% 0.21/0.47 inference(unit_resolution,[status(thm)],[119, 8])).
% 0.21/0.47 tff(121,plain,
% 0.21/0.47 (member(member_of_1_not_of_2(c, a), c)),
% 0.21/0.47 inference(unit_resolution,[status(thm)],[120, 116])).
% 0.21/0.47 tff(122,plain,
% 0.21/0.47 (((~![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))) | ((~equal_sets(complement(b), c)) | subset(c, complement(b)))) <=> ((~![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))) | (~equal_sets(complement(b), c)) | subset(c, complement(b)))),
% 0.21/0.47 inference(rewrite,[status(thm)],[])).
% 0.21/0.47 tff(123,plain,
% 0.21/0.47 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))) | ((~equal_sets(complement(b), c)) | subset(c, complement(b)))),
% 0.21/0.48 inference(quant_inst,[status(thm)],[])).
% 0.21/0.48 tff(124,plain,
% 0.21/0.48 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))) | (~equal_sets(complement(b), c)) | subset(c, complement(b))),
% 0.21/0.48 inference(modus_ponens,[status(thm)],[123, 122])).
% 0.21/0.48 tff(125,plain,
% 0.21/0.48 (subset(c, complement(b))),
% 0.21/0.48 inference(unit_resolution,[status(thm)],[124, 74, 28])).
% 0.21/0.48 tff(126,plain,
% 0.21/0.48 (((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(c, a), complement(b)) | (~member(member_of_1_not_of_2(c, a), c)) | (~subset(c, complement(b))))) <=> ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | member(member_of_1_not_of_2(c, a), complement(b)) | (~member(member_of_1_not_of_2(c, a), c)) | (~subset(c, complement(b))))),
% 0.21/0.48 inference(rewrite,[status(thm)],[])).
% 0.21/0.48 tff(127,plain,
% 0.21/0.48 ((member(member_of_1_not_of_2(c, a), complement(b)) | (~subset(c, complement(b))) | (~member(member_of_1_not_of_2(c, a), c))) <=> (member(member_of_1_not_of_2(c, a), complement(b)) | (~member(member_of_1_not_of_2(c, a), c)) | (~subset(c, complement(b))))),
% 0.21/0.48 inference(rewrite,[status(thm)],[])).
% 0.21/0.48 tff(128,plain,
% 0.21/0.48 (((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(c, a), complement(b)) | (~subset(c, complement(b))) | (~member(member_of_1_not_of_2(c, a), c)))) <=> ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(c, a), complement(b)) | (~member(member_of_1_not_of_2(c, a), c)) | (~subset(c, complement(b)))))),
% 0.21/0.48 inference(monotonicity,[status(thm)],[127])).
% 0.21/0.48 tff(129,plain,
% 0.21/0.48 (((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(c, a), complement(b)) | (~subset(c, complement(b))) | (~member(member_of_1_not_of_2(c, a), c)))) <=> ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | member(member_of_1_not_of_2(c, a), complement(b)) | (~member(member_of_1_not_of_2(c, a), c)) | (~subset(c, complement(b))))),
% 0.21/0.48 inference(transitivity,[status(thm)],[128, 126])).
% 0.21/0.48 tff(130,plain,
% 0.21/0.48 ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(c, a), complement(b)) | (~subset(c, complement(b))) | (~member(member_of_1_not_of_2(c, a), c)))),
% 0.21/0.48 inference(quant_inst,[status(thm)],[])).
% 0.21/0.48 tff(131,plain,
% 0.21/0.48 ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | member(member_of_1_not_of_2(c, a), complement(b)) | (~member(member_of_1_not_of_2(c, a), c)) | (~subset(c, complement(b)))),
% 0.21/0.48 inference(modus_ponens,[status(thm)],[130, 129])).
% 0.21/0.48 tff(132,plain,
% 0.21/0.48 (member(member_of_1_not_of_2(c, a), complement(b))),
% 0.21/0.48 inference(unit_resolution,[status(thm)],[131, 49, 125, 121])).
% 0.21/0.48 tff(133,plain,
% 0.21/0.48 (((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | ((~member(member_of_1_not_of_2(c, a), b)) | (~member(member_of_1_not_of_2(c, a), complement(b))))) <=> ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | (~member(member_of_1_not_of_2(c, a), b)) | (~member(member_of_1_not_of_2(c, a), complement(b))))),
% 0.21/0.48 inference(rewrite,[status(thm)],[])).
% 0.21/0.48 tff(134,plain,
% 0.21/0.48 ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | ((~member(member_of_1_not_of_2(c, a), b)) | (~member(member_of_1_not_of_2(c, a), complement(b))))),
% 0.21/0.48 inference(quant_inst,[status(thm)],[])).
% 0.21/0.48 tff(135,plain,
% 0.21/0.48 ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | (~member(member_of_1_not_of_2(c, a), b)) | (~member(member_of_1_not_of_2(c, a), complement(b)))),
% 0.21/0.48 inference(modus_ponens,[status(thm)],[134, 133])).
% 0.21/0.48 tff(136,plain,
% 0.21/0.48 (~member(member_of_1_not_of_2(c, a), b)),
% 0.21/0.48 inference(unit_resolution,[status(thm)],[135, 93, 132])).
% 0.21/0.48 tff(137,plain,
% 0.21/0.48 (((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(c, a), a)) | subset(c, 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(c, a), a)) | subset(c, a))),
% 0.21/0.48 inference(rewrite,[status(thm)],[])).
% 0.21/0.48 tff(138,plain,
% 0.21/0.48 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(c, a), a)) | subset(c, a))),
% 0.21/0.48 inference(quant_inst,[status(thm)],[])).
% 0.21/0.48 tff(139,plain,
% 0.21/0.48 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(c, a), a)) | subset(c, a)),
% 0.21/0.48 inference(modus_ponens,[status(thm)],[138, 137])).
% 0.21/0.48 tff(140,plain,
% 0.21/0.48 (~member(member_of_1_not_of_2(c, a), a)),
% 0.21/0.48 inference(unit_resolution,[status(thm)],[139, 20, 116])).
% 0.21/0.48 tff(141,plain,
% 0.21/0.48 (((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | (member(member_of_1_not_of_2(c, a), a) | member(member_of_1_not_of_2(c, a), complement(a)))) <=> ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | member(member_of_1_not_of_2(c, a), a) | member(member_of_1_not_of_2(c, a), complement(a)))),
% 0.21/0.48 inference(rewrite,[status(thm)],[])).
% 0.21/0.48 tff(142,plain,
% 0.21/0.48 ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | (member(member_of_1_not_of_2(c, a), a) | member(member_of_1_not_of_2(c, a), complement(a)))),
% 0.21/0.48 inference(quant_inst,[status(thm)],[])).
% 0.21/0.48 tff(143,plain,
% 0.21/0.48 ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | member(member_of_1_not_of_2(c, a), a) | member(member_of_1_not_of_2(c, a), complement(a))),
% 0.21/0.48 inference(modus_ponens,[status(thm)],[142, 141])).
% 0.21/0.48 tff(144,plain,
% 0.21/0.48 (member(member_of_1_not_of_2(c, a), complement(a))),
% 0.21/0.48 inference(unit_resolution,[status(thm)],[143, 60, 140])).
% 0.21/0.48 tff(145,plain,
% 0.21/0.48 (((~![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))) | ((~equal_sets(complement(a), b)) | subset(complement(a), b))) <=> ((~![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))) | (~equal_sets(complement(a), b)) | subset(complement(a), b))),
% 0.21/0.48 inference(rewrite,[status(thm)],[])).
% 0.21/0.48 tff(146,plain,
% 0.21/0.48 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))) | ((~equal_sets(complement(a), b)) | subset(complement(a), b))),
% 0.21/0.48 inference(quant_inst,[status(thm)],[])).
% 0.21/0.48 tff(147,plain,
% 0.21/0.48 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))) | (~equal_sets(complement(a), b)) | subset(complement(a), b)),
% 0.21/0.48 inference(modus_ponens,[status(thm)],[146, 145])).
% 0.21/0.48 tff(148,plain,
% 0.21/0.48 (subset(complement(a), b)),
% 0.21/0.48 inference(unit_resolution,[status(thm)],[147, 35, 67])).
% 0.21/0.48 tff(149,plain,
% 0.21/0.48 (((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | ((~subset(complement(a), b)) | (~member(member_of_1_not_of_2(c, a), complement(a))) | member(member_of_1_not_of_2(c, a), b))) <=> ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (~subset(complement(a), b)) | (~member(member_of_1_not_of_2(c, a), complement(a))) | member(member_of_1_not_of_2(c, a), b))),
% 0.21/0.48 inference(rewrite,[status(thm)],[])).
% 0.21/0.48 tff(150,plain,
% 0.21/0.48 ((member(member_of_1_not_of_2(c, a), b) | (~subset(complement(a), b)) | (~member(member_of_1_not_of_2(c, a), complement(a)))) <=> ((~subset(complement(a), b)) | (~member(member_of_1_not_of_2(c, a), complement(a))) | member(member_of_1_not_of_2(c, a), b))),
% 0.21/0.48 inference(rewrite,[status(thm)],[])).
% 0.21/0.48 tff(151,plain,
% 0.21/0.48 (((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(c, a), b) | (~subset(complement(a), b)) | (~member(member_of_1_not_of_2(c, a), complement(a))))) <=> ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | ((~subset(complement(a), b)) | (~member(member_of_1_not_of_2(c, a), complement(a))) | member(member_of_1_not_of_2(c, a), b)))),
% 0.21/0.48 inference(monotonicity,[status(thm)],[150])).
% 0.21/0.48 tff(152,plain,
% 0.21/0.48 (((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(c, a), b) | (~subset(complement(a), b)) | (~member(member_of_1_not_of_2(c, a), complement(a))))) <=> ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (~subset(complement(a), b)) | (~member(member_of_1_not_of_2(c, a), complement(a))) | member(member_of_1_not_of_2(c, a), b))),
% 0.21/0.48 inference(transitivity,[status(thm)],[151, 149])).
% 0.21/0.48 tff(153,plain,
% 0.21/0.48 ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (member(member_of_1_not_of_2(c, a), b) | (~subset(complement(a), b)) | (~member(member_of_1_not_of_2(c, a), complement(a))))),
% 0.21/0.48 inference(quant_inst,[status(thm)],[])).
% 0.21/0.48 tff(154,plain,
% 0.21/0.48 ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (~subset(complement(a), b)) | (~member(member_of_1_not_of_2(c, a), complement(a))) | member(member_of_1_not_of_2(c, a), b)),
% 0.21/0.48 inference(modus_ponens,[status(thm)],[153, 152])).
% 0.21/0.48 tff(155,plain,
% 0.21/0.48 ($false),
% 0.21/0.48 inference(unit_resolution,[status(thm)],[154, 49, 148, 144, 136])).
% 0.21/0.48 % SZS output end Proof
%------------------------------------------------------------------------------