%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET012-2 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n012.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 05:24:46 AM UTC 2025
% Result : Unsatisfiable 0.39s 0.59s
% Output : Proof 0.39s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : SET012-2 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.06/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n012.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 04:00:11 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.39/0.59 % SZS status Unsatisfiable
% 0.39/0.59 % SZS output start Proof
% 0.39/0.59 tff(member_type, type, (
% 0.39/0.59 member: ( $i * $i ) > $o)).
% 0.39/0.59 tff(complement_type, type, (
% 0.39/0.59 complement: $i > $i)).
% 0.39/0.59 tff(b_type, type, (
% 0.39/0.59 b: $i)).
% 0.39/0.59 tff(member_of_1_not_of_2_type, type, (
% 0.39/0.59 member_of_1_not_of_2: ( $i * $i ) > $i)).
% 0.39/0.59 tff(a_type, type, (
% 0.39/0.59 a: $i)).
% 0.39/0.59 tff(c_type, type, (
% 0.39/0.59 c: $i)).
% 0.39/0.59 tff(subset_type, type, (
% 0.39/0.59 subset: ( $i * $i ) > $o)).
% 0.39/0.59 tff(equal_sets_type, type, (
% 0.39/0.59 equal_sets: ( $i * $i ) > $o)).
% 0.39/0.59 tff(1,assumption,(~member(member_of_1_not_of_2(c, a), complement(b))), introduced(assumption)).
% 0.39/0.59 tff(2,assumption,(~subset(a, c)), introduced(assumption)).
% 0.39/0.59 tff(3,plain,
% 0.39/0.59 (^[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)))),
% 0.39/0.59 inference(bind,[status(th)],[])).
% 0.39/0.59 tff(4,plain,
% 0.39/0.59 (![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))),
% 0.39/0.59 inference(quant_intro,[status(thm)],[3])).
% 0.39/0.59 tff(5,plain,
% 0.39/0.59 (![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))),
% 0.39/0.59 inference(rewrite,[status(thm)],[])).
% 0.39/0.59 tff(6,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')).
% 0.39/0.59 tff(7,plain,
% 0.39/0.59 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.39/0.59 inference(modus_ponens,[status(thm)],[6, 5])).
% 0.39/0.59 tff(8,plain,(
% 0.39/0.59 ![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.39/0.60 inference(skolemize,[status(sab)],[7])).
% 0.39/0.60 tff(9,plain,
% 0.39/0.60 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[8, 4])).
% 0.39/0.60 tff(10,plain,
% 0.39/0.60 (((~![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))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(11,plain,
% 0.39/0.60 ((~![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))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(12,plain,
% 0.39/0.60 ((~![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)),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.39/0.60 tff(13,plain,
% 0.39/0.60 ((~member(member_of_1_not_of_2(a, c), c)) | subset(a, c)),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[12, 9])).
% 0.39/0.60 tff(14,plain,
% 0.39/0.60 (~member(member_of_1_not_of_2(a, c), c)),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[13, 2])).
% 0.39/0.60 tff(15,plain,
% 0.39/0.60 (equal_sets(complement(b), c) <=> equal_sets(complement(b), c)),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(16,axiom,(equal_sets(complement(b), c)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','complement_of_b_is_c')).
% 0.39/0.60 tff(17,plain,
% 0.39/0.60 (equal_sets(complement(b), c)),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.39/0.60 tff(18,plain,
% 0.39/0.60 (^[Subset: $i, Superset: $i] : refl(((~equal_sets(Subset, Superset)) | subset(Subset, Superset)) <=> ((~equal_sets(Subset, Superset)) | subset(Subset, Superset)))),
% 0.39/0.60 inference(bind,[status(th)],[])).
% 0.39/0.60 tff(19,plain,
% 0.39/0.60 (![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset)) <=> ![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))),
% 0.39/0.60 inference(quant_intro,[status(thm)],[18])).
% 0.39/0.60 tff(20,plain,
% 0.39/0.60 (![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset)) <=> ![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(21,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')).
% 0.39/0.60 tff(22,plain,
% 0.39/0.60 (![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.39/0.60 tff(23,plain,(
% 0.39/0.60 ![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))),
% 0.39/0.60 inference(skolemize,[status(sab)],[22])).
% 0.39/0.60 tff(24,plain,
% 0.39/0.60 (![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[23, 19])).
% 0.39/0.60 tff(25,plain,
% 0.39/0.60 (((~![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))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(26,plain,
% 0.39/0.60 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))) | ((~equal_sets(complement(b), c)) | subset(complement(b), c))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(27,plain,
% 0.39/0.60 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))) | (~equal_sets(complement(b), c)) | subset(complement(b), c)),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.39/0.60 tff(28,plain,
% 0.39/0.60 (subset(complement(b), c)),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[27, 24, 17])).
% 0.39/0.60 tff(29,plain,
% 0.39/0.60 (^[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))))),
% 0.39/0.60 inference(bind,[status(th)],[])).
% 0.39/0.60 tff(30,plain,
% 0.39/0.60 (![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)))),
% 0.39/0.60 inference(quant_intro,[status(thm)],[29])).
% 0.39/0.60 tff(31,plain,
% 0.39/0.60 (![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)))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(32,plain,
% 0.39/0.60 (^[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))))),
% 0.39/0.60 inference(bind,[status(th)],[])).
% 0.39/0.60 tff(33,plain,
% 0.39/0.60 (![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)))),
% 0.39/0.60 inference(quant_intro,[status(thm)],[32])).
% 0.39/0.60 tff(34,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')).
% 0.39/0.60 tff(35,plain,
% 0.39/0.60 (![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.39/0.60 tff(36,plain,
% 0.39/0.60 (![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[35, 31])).
% 0.39/0.60 tff(37,plain,(
% 0.39/0.60 ![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))),
% 0.39/0.60 inference(skolemize,[status(sab)],[36])).
% 0.39/0.60 tff(38,plain,
% 0.39/0.60 (![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[37, 30])).
% 0.39/0.60 tff(39,plain,
% 0.39/0.60 (((~![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))))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(40,plain,
% 0.39/0.60 ((~![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))))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(41,plain,
% 0.39/0.60 ((~![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)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.39/0.60 tff(42,plain,
% 0.39/0.60 (~member(member_of_1_not_of_2(a, c), complement(b))),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[41, 38, 28, 14])).
% 0.39/0.60 tff(43,plain,
% 0.39/0.60 (^[Xs: $i, X: $i] : refl((member(X, Xs) | member(X, complement(Xs))) <=> (member(X, Xs) | member(X, complement(Xs))))),
% 0.39/0.60 inference(bind,[status(th)],[])).
% 0.39/0.60 tff(44,plain,
% 0.39/0.60 (![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs))) <=> ![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.39/0.60 inference(quant_intro,[status(thm)],[43])).
% 0.39/0.60 tff(45,plain,
% 0.39/0.60 (![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs))) <=> ![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(46,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')).
% 0.39/0.60 tff(47,plain,
% 0.39/0.60 (![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.39/0.60 tff(48,plain,(
% 0.39/0.60 ![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.39/0.60 inference(skolemize,[status(sab)],[47])).
% 0.39/0.60 tff(49,plain,
% 0.39/0.60 (![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[48, 44])).
% 0.39/0.60 tff(50,plain,
% 0.39/0.60 (((~![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)))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(51,plain,
% 0.39/0.60 ((~![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)))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(52,plain,
% 0.39/0.60 ((~![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))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.39/0.60 tff(53,plain,
% 0.39/0.60 (member(member_of_1_not_of_2(a, c), b)),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[52, 49, 42])).
% 0.39/0.60 tff(54,plain,
% 0.39/0.60 (^[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)))),
% 0.39/0.60 inference(bind,[status(th)],[])).
% 0.39/0.60 tff(55,plain,
% 0.39/0.60 (![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))),
% 0.39/0.60 inference(quant_intro,[status(thm)],[54])).
% 0.39/0.60 tff(56,plain,
% 0.39/0.60 (![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))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(57,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')).
% 0.39/0.60 tff(58,plain,
% 0.39/0.60 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.39/0.60 tff(59,plain,(
% 0.39/0.60 ![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.39/0.60 inference(skolemize,[status(sab)],[58])).
% 0.39/0.60 tff(60,plain,
% 0.39/0.60 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[59, 55])).
% 0.39/0.60 tff(61,plain,
% 0.39/0.60 (((~![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))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(62,plain,
% 0.39/0.60 ((~![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))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(63,plain,
% 0.39/0.60 ((~![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)),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.39/0.60 tff(64,plain,
% 0.39/0.60 (subset(a, c) | member(member_of_1_not_of_2(a, c), a)),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[63, 60])).
% 0.39/0.60 tff(65,plain,
% 0.39/0.60 (member(member_of_1_not_of_2(a, c), a)),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[64, 2])).
% 0.39/0.60 tff(66,assumption,(member(member_of_1_not_of_2(a, c), complement(a))), introduced(assumption)).
% 0.39/0.60 tff(67,assumption,(member(member_of_1_not_of_2(a, c), a)), introduced(assumption)).
% 0.39/0.60 tff(68,plain,
% 0.39/0.60 (^[Xs: $i, X: $i] : refl(((~member(X, Xs)) | (~member(X, complement(Xs)))) <=> ((~member(X, Xs)) | (~member(X, complement(Xs)))))),
% 0.39/0.60 inference(bind,[status(th)],[])).
% 0.39/0.60 tff(69,plain,
% 0.39/0.60 (![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs)))) <=> ![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.39/0.60 inference(quant_intro,[status(thm)],[68])).
% 0.39/0.60 tff(70,plain,
% 0.39/0.60 (![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs)))) <=> ![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(71,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')).
% 0.39/0.60 tff(72,plain,
% 0.39/0.60 (![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.39/0.60 tff(73,plain,(
% 0.39/0.60 ![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.39/0.60 inference(skolemize,[status(sab)],[72])).
% 0.39/0.60 tff(74,plain,
% 0.39/0.60 (![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[73, 69])).
% 0.39/0.60 tff(75,plain,
% 0.39/0.60 (((~![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))))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(76,plain,
% 0.39/0.60 ((~![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))))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(77,plain,
% 0.39/0.60 ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | (~member(member_of_1_not_of_2(a, c), a)) | (~member(member_of_1_not_of_2(a, c), complement(a)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.39/0.60 tff(78,plain,
% 0.39/0.60 ($false),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[77, 74, 67, 66])).
% 0.39/0.60 tff(79,plain,((~member(member_of_1_not_of_2(a, c), complement(a))) | (~member(member_of_1_not_of_2(a, c), a))), inference(lemma,lemma(discharge,[]))).
% 0.39/0.60 tff(80,plain,
% 0.39/0.60 (~member(member_of_1_not_of_2(a, c), complement(a))),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[79, 65])).
% 0.39/0.60 tff(81,plain,
% 0.39/0.60 (equal_sets(complement(a), b) <=> equal_sets(complement(a), b)),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(82,axiom,(equal_sets(complement(a), b)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','complement_of_a_is_b')).
% 0.39/0.60 tff(83,plain,
% 0.39/0.60 (equal_sets(complement(a), b)),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.39/0.60 tff(84,plain,
% 0.39/0.60 (^[Subset: $i, Superset: $i] : refl(((~equal_sets(Superset, Subset)) | subset(Subset, Superset)) <=> ((~equal_sets(Superset, Subset)) | subset(Subset, Superset)))),
% 0.39/0.60 inference(bind,[status(th)],[])).
% 0.39/0.60 tff(85,plain,
% 0.39/0.60 (![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset)) <=> ![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))),
% 0.39/0.60 inference(quant_intro,[status(thm)],[84])).
% 0.39/0.60 tff(86,plain,
% 0.39/0.60 (![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset)) <=> ![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(87,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')).
% 0.39/0.60 tff(88,plain,
% 0.39/0.60 (![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[87, 86])).
% 0.39/0.60 tff(89,plain,(
% 0.39/0.60 ![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))),
% 0.39/0.60 inference(skolemize,[status(sab)],[88])).
% 0.39/0.60 tff(90,plain,
% 0.39/0.60 (![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[89, 85])).
% 0.39/0.60 tff(91,plain,
% 0.39/0.60 (((~![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)))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(92,plain,
% 0.39/0.60 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))) | ((~equal_sets(complement(a), b)) | subset(b, complement(a)))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(93,plain,
% 0.39/0.60 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))) | (~equal_sets(complement(a), b)) | subset(b, complement(a))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[92, 91])).
% 0.39/0.60 tff(94,plain,
% 0.39/0.60 (subset(b, complement(a))),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[93, 90, 83])).
% 0.39/0.60 tff(95,plain,
% 0.39/0.60 (((~![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)) | (~member(member_of_1_not_of_2(a, c), b)) | (~subset(b, complement(a))))) <=> ((~![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)) | (~member(member_of_1_not_of_2(a, c), b)) | (~subset(b, complement(a))))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(96,plain,
% 0.39/0.60 ((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))) <=> (member(member_of_1_not_of_2(a, c), complement(a)) | (~member(member_of_1_not_of_2(a, c), b)) | (~subset(b, complement(a))))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(97,plain,
% 0.39/0.60 (((~![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)))) | (member(member_of_1_not_of_2(a, c), complement(a)) | (~member(member_of_1_not_of_2(a, c), b)) | (~subset(b, complement(a)))))),
% 0.39/0.60 inference(monotonicity,[status(thm)],[96])).
% 0.39/0.60 tff(98,plain,
% 0.39/0.60 (((~![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)))) | member(member_of_1_not_of_2(a, c), complement(a)) | (~member(member_of_1_not_of_2(a, c), b)) | (~subset(b, complement(a))))),
% 0.39/0.60 inference(transitivity,[status(thm)],[97, 95])).
% 0.39/0.60 tff(99,plain,
% 0.39/0.60 ((~![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)))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(100,plain,
% 0.39/0.60 ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | member(member_of_1_not_of_2(a, c), complement(a)) | (~member(member_of_1_not_of_2(a, c), b)) | (~subset(b, complement(a)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.39/0.60 tff(101,plain,
% 0.39/0.60 ($false),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[100, 38, 94, 80, 53])).
% 0.39/0.60 tff(102,plain,(subset(a, c)), inference(lemma,lemma(discharge,[]))).
% 0.39/0.60 tff(103,plain,
% 0.39/0.60 ((~equal_sets(a, c)) <=> (~equal_sets(a, c))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(104,axiom,(~equal_sets(a, c)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_a_equals_c')).
% 0.39/0.60 tff(105,plain,
% 0.39/0.60 (~equal_sets(a, c)),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[104, 103])).
% 0.39/0.60 tff(106,plain,
% 0.39/0.60 (^[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))))),
% 0.39/0.60 inference(bind,[status(th)],[])).
% 0.39/0.60 tff(107,plain,
% 0.39/0.60 (![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)))),
% 0.39/0.60 inference(quant_intro,[status(thm)],[106])).
% 0.39/0.60 tff(108,plain,
% 0.39/0.60 (![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)))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(109,plain,
% 0.39/0.60 (^[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)))))),
% 0.39/0.60 inference(bind,[status(th)],[])).
% 0.39/0.60 tff(110,plain,
% 0.39/0.60 (![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)))),
% 0.39/0.60 inference(quant_intro,[status(thm)],[109])).
% 0.39/0.60 tff(111,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')).
% 0.39/0.60 tff(112,plain,
% 0.39/0.60 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[111, 110])).
% 0.39/0.60 tff(113,plain,
% 0.39/0.60 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[112, 108])).
% 0.39/0.60 tff(114,plain,(
% 0.39/0.60 ![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.39/0.60 inference(skolemize,[status(sab)],[113])).
% 0.39/0.60 tff(115,plain,
% 0.39/0.60 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[114, 107])).
% 0.39/0.60 tff(116,plain,
% 0.39/0.60 (((~![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)))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(117,plain,
% 0.39/0.60 ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | (equal_sets(a, c) | (~subset(a, c)) | (~subset(c, a)))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(118,plain,
% 0.39/0.60 ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | equal_sets(a, c) | (~subset(a, c)) | (~subset(c, a))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[117, 116])).
% 0.39/0.60 tff(119,plain,
% 0.39/0.60 ((~subset(a, c)) | (~subset(c, a))),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[118, 115, 105])).
% 0.39/0.60 tff(120,plain,
% 0.39/0.60 (~subset(c, a)),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[119, 102])).
% 0.39/0.60 tff(121,plain,
% 0.39/0.60 (((~![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))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(122,plain,
% 0.39/0.60 ((~![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))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(123,plain,
% 0.39/0.60 ((~![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)),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[122, 121])).
% 0.39/0.60 tff(124,plain,
% 0.39/0.60 (member(member_of_1_not_of_2(c, a), c)),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[123, 60, 120])).
% 0.39/0.60 tff(125,plain,
% 0.39/0.60 (((~![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)))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(126,plain,
% 0.39/0.60 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))) | ((~equal_sets(complement(b), c)) | subset(c, complement(b)))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(127,plain,
% 0.39/0.60 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Superset, Subset)) | subset(Subset, Superset))) | (~equal_sets(complement(b), c)) | subset(c, complement(b))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[126, 125])).
% 0.39/0.60 tff(128,plain,
% 0.39/0.60 (subset(c, complement(b))),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[127, 90, 17])).
% 0.39/0.60 tff(129,plain,
% 0.39/0.60 (((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | ((~member(member_of_1_not_of_2(c, a), c)) | member(member_of_1_not_of_2(c, a), complement(b)) | (~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), c)) | member(member_of_1_not_of_2(c, a), complement(b)) | (~subset(c, complement(b))))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(130,plain,
% 0.39/0.60 ((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), c)) | member(member_of_1_not_of_2(c, a), complement(b)) | (~subset(c, complement(b))))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(131,plain,
% 0.39/0.60 (((~![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), c)) | member(member_of_1_not_of_2(c, a), complement(b)) | (~subset(c, complement(b)))))),
% 0.39/0.60 inference(monotonicity,[status(thm)],[130])).
% 0.39/0.60 tff(132,plain,
% 0.39/0.60 (((~![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), c)) | member(member_of_1_not_of_2(c, a), complement(b)) | (~subset(c, complement(b))))),
% 0.39/0.60 inference(transitivity,[status(thm)],[131, 129])).
% 0.39/0.60 tff(133,plain,
% 0.39/0.60 ((~![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)))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(134,plain,
% 0.39/0.60 ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | (~member(member_of_1_not_of_2(c, a), c)) | member(member_of_1_not_of_2(c, a), complement(b)) | (~subset(c, complement(b)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[133, 132])).
% 0.39/0.60 tff(135,plain,
% 0.39/0.60 ($false),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[134, 38, 128, 124, 1])).
% 0.39/0.60 tff(136,plain,(member(member_of_1_not_of_2(c, a), complement(b))), inference(lemma,lemma(discharge,[]))).
% 0.39/0.60 tff(137,assumption,(~member(member_of_1_not_of_2(c, a), b)), introduced(assumption)).
% 0.39/0.60 tff(138,assumption,(~member(member_of_1_not_of_2(c, a), complement(a))), introduced(assumption)).
% 0.39/0.60 tff(139,plain,
% 0.39/0.60 (((~![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))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(140,plain,
% 0.39/0.60 ((~![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))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(141,plain,
% 0.39/0.60 ((~![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)),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[140, 139])).
% 0.39/0.60 tff(142,plain,
% 0.39/0.60 (~member(member_of_1_not_of_2(c, a), a)),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[141, 9, 120])).
% 0.39/0.60 tff(143,plain,
% 0.39/0.60 (((~![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)))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(144,plain,
% 0.39/0.60 ((~![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)))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(145,plain,
% 0.39/0.60 ((~![Xs: $i, X: $i] : (member(X, Xs) | member(X, complement(Xs)))) | member(member_of_1_not_of_2(c, a), a) | member(member_of_1_not_of_2(c, a), complement(a))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[144, 143])).
% 0.39/0.60 tff(146,plain,
% 0.39/0.60 ($false),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[145, 49, 142, 138])).
% 0.39/0.60 tff(147,plain,(member(member_of_1_not_of_2(c, a), complement(a))), inference(lemma,lemma(discharge,[]))).
% 0.39/0.60 tff(148,plain,
% 0.39/0.60 (((~![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))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(149,plain,
% 0.39/0.60 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))) | ((~equal_sets(complement(a), b)) | subset(complement(a), b))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(150,plain,
% 0.39/0.60 ((~![Subset: $i, Superset: $i] : ((~equal_sets(Subset, Superset)) | subset(Subset, Superset))) | (~equal_sets(complement(a), b)) | subset(complement(a), b)),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[149, 148])).
% 0.39/0.60 tff(151,plain,
% 0.39/0.60 (subset(complement(a), b)),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[150, 24, 83])).
% 0.39/0.60 tff(152,plain,
% 0.39/0.60 (((~![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)))) | 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))))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(153,plain,
% 0.39/0.60 ((~![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))))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(154,plain,
% 0.39/0.60 ((~![Subset: $i, Element: $i, Superset: $i] : (member(Element, Superset) | (~subset(Subset, Superset)) | (~member(Element, Subset)))) | member(member_of_1_not_of_2(c, a), b) | (~subset(complement(a), b)) | (~member(member_of_1_not_of_2(c, a), complement(a)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[153, 152])).
% 0.39/0.60 tff(155,plain,
% 0.39/0.60 ($false),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[154, 38, 151, 147, 137])).
% 0.39/0.60 tff(156,plain,(member(member_of_1_not_of_2(c, a), b)), inference(lemma,lemma(discharge,[]))).
% 0.39/0.60 tff(157,plain,
% 0.39/0.60 (((~![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))))),
% 0.39/0.60 inference(rewrite,[status(thm)],[])).
% 0.39/0.60 tff(158,plain,
% 0.39/0.60 ((~![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))))),
% 0.39/0.60 inference(quant_inst,[status(thm)],[])).
% 0.39/0.60 tff(159,plain,
% 0.39/0.60 ((~![Xs: $i, X: $i] : ((~member(X, Xs)) | (~member(X, complement(Xs))))) | (~member(member_of_1_not_of_2(c, a), b)) | (~member(member_of_1_not_of_2(c, a), complement(b)))),
% 0.39/0.60 inference(modus_ponens,[status(thm)],[158, 157])).
% 0.39/0.60 tff(160,plain,
% 0.39/0.60 ($false),
% 0.39/0.60 inference(unit_resolution,[status(thm)],[159, 74, 156, 136])).
% 0.39/0.60 % SZS output end Proof
% 0.39/0.61 % E exiting
%------------------------------------------------------------------------------