%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU010+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n016.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 05:26:47 AM UTC 2025
% Result : Theorem 1.24s 1.40s
% Output : Proof 1.24s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SEU010+1 : TPTP v9.0.0. Released v3.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n016.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 : Fri Jun 20 05:57:02 EDT 2025
% 0.12/0.34 % CPUTime :
% 1.24/1.40 % SZS status Theorem
% 1.24/1.40 % SZS output start Proof
% 1.24/1.40 tff(subset_type, type, (
% 1.24/1.40 subset: ( $i * $i ) > $o)).
% 1.24/1.40 tff(relation_dom_type, type, (
% 1.24/1.40 relation_dom: $i > $i)).
% 1.24/1.40 tff(tptp_fun_A_9_type, type, (
% 1.24/1.40 tptp_fun_A_9: $i)).
% 1.24/1.40 tff(relation_type, type, (
% 1.24/1.40 relation: $i > $o)).
% 1.24/1.40 tff(relation_composition_type, type, (
% 1.24/1.40 relation_composition: ( $i * $i ) > $i)).
% 1.24/1.40 tff(identity_relation_type, type, (
% 1.24/1.40 identity_relation: $i > $i)).
% 1.24/1.40 tff(relation_rng_type, type, (
% 1.24/1.40 relation_rng: $i > $i)).
% 1.24/1.40 tff(function_type, type, (
% 1.24/1.40 function: $i > $o)).
% 1.24/1.40 tff(1,plain,
% 1.24/1.40 (^[A: $i, B: $i] : refl(((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B))) <=> ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B))))),
% 1.24/1.40 inference(bind,[status(th)],[])).
% 1.24/1.40 tff(2,plain,
% 1.24/1.40 (![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B))) <=> ![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))),
% 1.24/1.40 inference(quant_intro,[status(thm)],[1])).
% 1.24/1.40 tff(3,plain,
% 1.24/1.40 (![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B))) <=> ![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))),
% 1.24/1.40 inference(rewrite,[status(thm)],[])).
% 1.24/1.40 tff(4,plain,
% 1.24/1.40 (^[A: $i, B: $i] : trans(monotonicity(rewrite((subset(relation_dom(B), A) => (relation_composition(identity_relation(A), B) = B)) <=> ((~subset(relation_dom(B), A)) | (relation_composition(identity_relation(A), B) = B))), ((relation(B) => (subset(relation_dom(B), A) => (relation_composition(identity_relation(A), B) = B))) <=> (relation(B) => ((~subset(relation_dom(B), A)) | (relation_composition(identity_relation(A), B) = B))))), rewrite((relation(B) => ((~subset(relation_dom(B), A)) | (relation_composition(identity_relation(A), B) = B))) <=> ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))), ((relation(B) => (subset(relation_dom(B), A) => (relation_composition(identity_relation(A), B) = B))) <=> ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))))),
% 1.24/1.40 inference(bind,[status(th)],[])).
% 1.24/1.40 tff(5,plain,
% 1.24/1.40 (![A: $i, B: $i] : (relation(B) => (subset(relation_dom(B), A) => (relation_composition(identity_relation(A), B) = B))) <=> ![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))),
% 1.24/1.40 inference(quant_intro,[status(thm)],[4])).
% 1.24/1.40 tff(6,axiom,(![A: $i, B: $i] : (relation(B) => (subset(relation_dom(B), A) => (relation_composition(identity_relation(A), B) = B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t77_relat_1')).
% 1.24/1.40 tff(7,plain,
% 1.24/1.40 (![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[6, 5])).
% 1.24/1.40 tff(8,plain,
% 1.24/1.40 (![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[7, 3])).
% 1.24/1.40 tff(9,plain,(
% 1.24/1.40 ![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))),
% 1.24/1.40 inference(skolemize,[status(sab)],[8])).
% 1.24/1.40 tff(10,plain,
% 1.24/1.40 (![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[9, 2])).
% 1.24/1.40 tff(11,assumption,(~subset(relation_rng(A!9), relation_rng(A!9))), introduced(assumption)).
% 1.24/1.40 tff(12,plain,
% 1.24/1.40 (^[A: $i] : refl(subset(A, A) <=> subset(A, A))),
% 1.24/1.40 inference(bind,[status(th)],[])).
% 1.24/1.40 tff(13,plain,
% 1.24/1.40 (![A: $i] : subset(A, A) <=> ![A: $i] : subset(A, A)),
% 1.24/1.40 inference(quant_intro,[status(thm)],[12])).
% 1.24/1.40 tff(14,plain,
% 1.24/1.40 (![A: $i] : subset(A, A) <=> ![A: $i] : subset(A, A)),
% 1.24/1.40 inference(rewrite,[status(thm)],[])).
% 1.24/1.40 tff(15,plain,
% 1.24/1.40 (![A: $i, B: $i] : subset(A, A) <=> ![A: $i] : subset(A, A)),
% 1.24/1.40 inference(elim_unused_vars,[status(thm)],[])).
% 1.24/1.40 tff(16,axiom,(![A: $i, B: $i] : subset(A, A)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','reflexivity_r1_tarski')).
% 1.24/1.40 tff(17,plain,
% 1.24/1.40 (![A: $i] : subset(A, A)),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[16, 15])).
% 1.24/1.40 tff(18,plain,
% 1.24/1.40 (![A: $i] : subset(A, A)),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[17, 14])).
% 1.24/1.40 tff(19,plain,(
% 1.24/1.40 ![A: $i] : subset(A, A)),
% 1.24/1.40 inference(skolemize,[status(sab)],[18])).
% 1.24/1.40 tff(20,plain,
% 1.24/1.40 (![A: $i] : subset(A, A)),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[19, 13])).
% 1.24/1.40 tff(21,plain,
% 1.24/1.40 ((~![A: $i] : subset(A, A)) | subset(relation_rng(A!9), relation_rng(A!9))),
% 1.24/1.40 inference(quant_inst,[status(thm)],[])).
% 1.24/1.40 tff(22,plain,
% 1.24/1.40 ($false),
% 1.24/1.40 inference(unit_resolution,[status(thm)],[21, 20, 11])).
% 1.24/1.40 tff(23,plain,(subset(relation_rng(A!9), relation_rng(A!9))), inference(lemma,lemma(discharge,[]))).
% 1.24/1.40 tff(24,plain,
% 1.24/1.40 (^[A: $i, B: $i] : refl(((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A))) <=> ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A))))),
% 1.24/1.40 inference(bind,[status(th)],[])).
% 1.24/1.40 tff(25,plain,
% 1.24/1.40 (![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A))) <=> ![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))),
% 1.24/1.40 inference(quant_intro,[status(thm)],[24])).
% 1.24/1.40 tff(26,plain,
% 1.24/1.40 (![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A))) <=> ![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))),
% 1.24/1.40 inference(rewrite,[status(thm)],[])).
% 1.24/1.40 tff(27,plain,
% 1.24/1.40 (^[A: $i, B: $i] : trans(monotonicity(rewrite((subset(relation_rng(B), A) => (relation_composition(B, identity_relation(A)) = B)) <=> ((~subset(relation_rng(B), A)) | (relation_composition(B, identity_relation(A)) = B))), ((relation(B) => (subset(relation_rng(B), A) => (relation_composition(B, identity_relation(A)) = B))) <=> (relation(B) => ((~subset(relation_rng(B), A)) | (relation_composition(B, identity_relation(A)) = B))))), rewrite((relation(B) => ((~subset(relation_rng(B), A)) | (relation_composition(B, identity_relation(A)) = B))) <=> ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))), ((relation(B) => (subset(relation_rng(B), A) => (relation_composition(B, identity_relation(A)) = B))) <=> ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))))),
% 1.24/1.40 inference(bind,[status(th)],[])).
% 1.24/1.40 tff(28,plain,
% 1.24/1.40 (![A: $i, B: $i] : (relation(B) => (subset(relation_rng(B), A) => (relation_composition(B, identity_relation(A)) = B))) <=> ![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))),
% 1.24/1.40 inference(quant_intro,[status(thm)],[27])).
% 1.24/1.40 tff(29,axiom,(![A: $i, B: $i] : (relation(B) => (subset(relation_rng(B), A) => (relation_composition(B, identity_relation(A)) = B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t79_relat_1')).
% 1.24/1.40 tff(30,plain,
% 1.24/1.40 (![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[29, 28])).
% 1.24/1.40 tff(31,plain,
% 1.24/1.40 (![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[30, 26])).
% 1.24/1.40 tff(32,plain,(
% 1.24/1.40 ![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))),
% 1.24/1.40 inference(skolemize,[status(sab)],[31])).
% 1.24/1.40 tff(33,plain,
% 1.24/1.40 (![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[32, 25])).
% 1.24/1.40 tff(34,plain,
% 1.24/1.40 ((~![A: $i] : ((~(relation(A) & function(A))) | ((relation_composition(identity_relation(relation_dom(A)), A) = A) & (relation_composition(A, identity_relation(relation_rng(A))) = A)))) <=> (~![A: $i] : ((~(relation(A) & function(A))) | ((relation_composition(identity_relation(relation_dom(A)), A) = A) & (relation_composition(A, identity_relation(relation_rng(A))) = A))))),
% 1.24/1.40 inference(rewrite,[status(thm)],[])).
% 1.24/1.40 tff(35,plain,
% 1.24/1.40 ((~![A: $i] : ((relation(A) & function(A)) => ((relation_composition(identity_relation(relation_dom(A)), A) = A) & (relation_composition(A, identity_relation(relation_rng(A))) = A)))) <=> (~![A: $i] : ((~(relation(A) & function(A))) | ((relation_composition(identity_relation(relation_dom(A)), A) = A) & (relation_composition(A, identity_relation(relation_rng(A))) = A))))),
% 1.24/1.40 inference(rewrite,[status(thm)],[])).
% 1.24/1.40 tff(36,axiom,(~![A: $i] : ((relation(A) & function(A)) => ((relation_composition(identity_relation(relation_dom(A)), A) = A) & (relation_composition(A, identity_relation(relation_rng(A))) = A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t42_funct_1')).
% 1.24/1.40 tff(37,plain,
% 1.24/1.40 (~![A: $i] : ((~(relation(A) & function(A))) | ((relation_composition(identity_relation(relation_dom(A)), A) = A) & (relation_composition(A, identity_relation(relation_rng(A))) = A)))),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[36, 35])).
% 1.24/1.40 tff(38,plain,
% 1.24/1.40 (~![A: $i] : ((~(relation(A) & function(A))) | ((relation_composition(identity_relation(relation_dom(A)), A) = A) & (relation_composition(A, identity_relation(relation_rng(A))) = A)))),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[37, 34])).
% 1.24/1.40 tff(39,plain,
% 1.24/1.40 (~![A: $i] : ((~(relation(A) & function(A))) | ((relation_composition(identity_relation(relation_dom(A)), A) = A) & (relation_composition(A, identity_relation(relation_rng(A))) = A)))),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[38, 34])).
% 1.24/1.40 tff(40,plain,
% 1.24/1.40 (~![A: $i] : ((~(relation(A) & function(A))) | ((relation_composition(identity_relation(relation_dom(A)), A) = A) & (relation_composition(A, identity_relation(relation_rng(A))) = A)))),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[39, 34])).
% 1.24/1.40 tff(41,plain,(
% 1.24/1.40 ~((~(relation(A!9) & function(A!9))) | ((relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) & (relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9)))),
% 1.24/1.40 inference(skolemize,[status(sab)],[40])).
% 1.24/1.40 tff(42,plain,
% 1.24/1.40 (relation(A!9) & function(A!9)),
% 1.24/1.40 inference(or_elim,[status(thm)],[41])).
% 1.24/1.40 tff(43,plain,
% 1.24/1.40 (relation(A!9)),
% 1.24/1.40 inference(and_elim,[status(thm)],[42])).
% 1.24/1.40 tff(44,plain,
% 1.24/1.40 (((~![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))) | ((relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9) | (~relation(A!9)) | (~subset(relation_rng(A!9), relation_rng(A!9))))) <=> ((~![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))) | (relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9) | (~relation(A!9)) | (~subset(relation_rng(A!9), relation_rng(A!9))))),
% 1.24/1.40 inference(rewrite,[status(thm)],[])).
% 1.24/1.40 tff(45,plain,
% 1.24/1.40 ((~![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))) | ((relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9) | (~relation(A!9)) | (~subset(relation_rng(A!9), relation_rng(A!9))))),
% 1.24/1.40 inference(quant_inst,[status(thm)],[])).
% 1.24/1.40 tff(46,plain,
% 1.24/1.40 ((~![A: $i, B: $i] : ((relation_composition(B, identity_relation(A)) = B) | (~relation(B)) | (~subset(relation_rng(B), A)))) | (relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9) | (~relation(A!9)) | (~subset(relation_rng(A!9), relation_rng(A!9)))),
% 1.24/1.40 inference(modus_ponens,[status(thm)],[45, 44])).
% 1.24/1.40 tff(47,plain,
% 1.24/1.40 (relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9),
% 1.24/1.40 inference(unit_resolution,[status(thm)],[46, 43, 33, 23])).
% 1.24/1.40 tff(48,plain,
% 1.24/1.40 ((~(~((~(relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9)) | (~(relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9))))) <=> ((~(relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9)) | (~(relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9)))),
% 1.24/1.41 inference(rewrite,[status(thm)],[])).
% 1.24/1.41 tff(49,plain,
% 1.24/1.41 (((relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) & (relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9)) <=> (~((~(relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9)) | (~(relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9))))),
% 1.24/1.41 inference(rewrite,[status(thm)],[])).
% 1.24/1.41 tff(50,plain,
% 1.24/1.41 ((~((relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) & (relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9))) <=> (~(~((~(relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9)) | (~(relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9)))))),
% 1.24/1.41 inference(monotonicity,[status(thm)],[49])).
% 1.24/1.41 tff(51,plain,
% 1.24/1.41 ((~((relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) & (relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9))) <=> ((~(relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9)) | (~(relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9)))),
% 1.24/1.41 inference(transitivity,[status(thm)],[50, 48])).
% 1.24/1.41 tff(52,plain,
% 1.24/1.41 (~((relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) & (relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9))),
% 1.24/1.41 inference(or_elim,[status(thm)],[41])).
% 1.24/1.41 tff(53,plain,
% 1.24/1.41 ((~(relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9)) | (~(relation_composition(A!9, identity_relation(relation_rng(A!9))) = A!9))),
% 1.24/1.41 inference(modus_ponens,[status(thm)],[52, 51])).
% 1.24/1.41 tff(54,plain,
% 1.24/1.41 (~(relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9)),
% 1.24/1.41 inference(unit_resolution,[status(thm)],[53, 47])).
% 1.24/1.41 tff(55,plain,
% 1.24/1.41 (((~![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))) | ((relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) | (~relation(A!9)) | (~subset(relation_dom(A!9), relation_dom(A!9))))) <=> ((~![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))) | (relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) | (~relation(A!9)) | (~subset(relation_dom(A!9), relation_dom(A!9))))),
% 1.24/1.41 inference(rewrite,[status(thm)],[])).
% 1.24/1.41 tff(56,plain,
% 1.24/1.41 (((relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) | (~subset(relation_dom(A!9), relation_dom(A!9))) | (~relation(A!9))) <=> ((relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) | (~relation(A!9)) | (~subset(relation_dom(A!9), relation_dom(A!9))))),
% 1.24/1.41 inference(rewrite,[status(thm)],[])).
% 1.24/1.41 tff(57,plain,
% 1.24/1.41 (((~![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))) | ((relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) | (~subset(relation_dom(A!9), relation_dom(A!9))) | (~relation(A!9)))) <=> ((~![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))) | ((relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) | (~relation(A!9)) | (~subset(relation_dom(A!9), relation_dom(A!9)))))),
% 1.24/1.41 inference(monotonicity,[status(thm)],[56])).
% 1.24/1.41 tff(58,plain,
% 1.24/1.41 (((~![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))) | ((relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) | (~subset(relation_dom(A!9), relation_dom(A!9))) | (~relation(A!9)))) <=> ((~![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))) | (relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) | (~relation(A!9)) | (~subset(relation_dom(A!9), relation_dom(A!9))))),
% 1.24/1.41 inference(transitivity,[status(thm)],[57, 55])).
% 1.24/1.41 tff(59,plain,
% 1.24/1.41 ((~![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))) | ((relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) | (~subset(relation_dom(A!9), relation_dom(A!9))) | (~relation(A!9)))),
% 1.24/1.41 inference(quant_inst,[status(thm)],[])).
% 1.24/1.41 tff(60,plain,
% 1.24/1.41 ((~![A: $i, B: $i] : ((relation_composition(identity_relation(A), B) = B) | (~subset(relation_dom(B), A)) | (~relation(B)))) | (relation_composition(identity_relation(relation_dom(A!9)), A!9) = A!9) | (~relation(A!9)) | (~subset(relation_dom(A!9), relation_dom(A!9)))),
% 1.24/1.41 inference(modus_ponens,[status(thm)],[59, 58])).
% 1.24/1.41 tff(61,plain,
% 1.24/1.41 (~subset(relation_dom(A!9), relation_dom(A!9))),
% 1.24/1.41 inference(unit_resolution,[status(thm)],[60, 43, 54, 10])).
% 1.24/1.41 tff(62,plain,
% 1.24/1.41 ((~![A: $i] : subset(A, A)) | subset(relation_dom(A!9), relation_dom(A!9))),
% 1.24/1.41 inference(quant_inst,[status(thm)],[])).
% 1.24/1.41 tff(63,plain,
% 1.24/1.41 ($false),
% 1.24/1.41 inference(unit_resolution,[status(thm)],[62, 20, 61])).
% 1.24/1.41 % SZS output end Proof
% 1.24/1.41 % E exiting
%------------------------------------------------------------------------------