TSTP Solution File: SET666+3 by SuperZenon---0.0.1
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- Process Solution
%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : SET666+3 : TPTP v8.1.0. Released v2.2.0.
% Transfm : none
% Format : tptp:raw
% Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% Computer : n009.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 : 600s
% DateTime : Tue Jul 19 05:43:14 EDT 2022
% Result : Theorem 9.23s 9.40s
% Output : Proof 9.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET666+3 : TPTP v8.1.0. Released v2.2.0.
% 0.03/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.13/0.34 % Computer : n009.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 600
% 0.13/0.34 % DateTime : Sun Jul 10 17:09:07 EDT 2022
% 0.13/0.34 % CPUTime :
% 9.23/9.40 % SZS status Theorem
% 9.23/9.40 (* PROOF-FOUND *)
% 9.23/9.40 (* BEGIN-PROOF *)
% 9.23/9.40 % SZS output start Proof
% 9.23/9.40 1. (ilf_type T_0 (set_type)) (-. (ilf_type T_0 (set_type))) ### Axiom
% 9.23/9.40 2. (-. (ilf_type (identity_relation_of T_0) (set_type))) (ilf_type (identity_relation_of T_0) (set_type)) ### Axiom
% 9.23/9.40 3. (All B, (ilf_type B (set_type))) (-. (ilf_type (identity_relation_of T_0) (set_type))) ### All 2
% 9.23/9.40 4. (ilf_type T_0 (set_type)) (-. (ilf_type T_0 (set_type))) ### Axiom
% 9.23/9.40 5. (ilf_type T_0 (set_type)) (-. (ilf_type T_0 (set_type))) ### Axiom
% 9.23/9.40 6. (ilf_type T_0 (set_type)) (-. (ilf_type T_0 (set_type))) ### Axiom
% 9.23/9.40 7. (ilf_type T_0 (set_type)) (-. (ilf_type T_0 (set_type))) ### Axiom
% 9.23/9.40 8. (-. (ilf_type (cross_product T_0 T_0) (set_type))) (ilf_type (cross_product T_0 T_0) (set_type)) ### Axiom
% 9.23/9.40 9. ((ilf_type T_0 (set_type)) => (ilf_type (cross_product T_0 T_0) (set_type))) (-. (ilf_type (cross_product T_0 T_0) (set_type))) (ilf_type T_0 (set_type)) ### Imply 7 8
% 9.23/9.40 10. (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product T_0 C) (set_type)))) (ilf_type T_0 (set_type)) (-. (ilf_type (cross_product T_0 T_0) (set_type))) ### All 9
% 9.23/9.40 11. ((ilf_type T_0 (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product T_0 C) (set_type))))) (-. (ilf_type (cross_product T_0 T_0) (set_type))) (ilf_type T_0 (set_type)) ### Imply 6 10
% 9.23/9.40 12. (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (-. (ilf_type (cross_product T_0 T_0) (set_type))) ### All 11
% 9.23/9.40 13. (-. (ilf_type (power_set (cross_product T_0 T_0)) (set_type))) (ilf_type (power_set (cross_product T_0 T_0)) (set_type)) ### Axiom
% 9.23/9.40 14. ((-. (empty (power_set (cross_product T_0 T_0)))) /\ (ilf_type (power_set (cross_product T_0 T_0)) (set_type))) (-. (ilf_type (power_set (cross_product T_0 T_0)) (set_type))) ### And 13
% 9.23/9.40 15. ((ilf_type (cross_product T_0 T_0) (set_type)) => ((-. (empty (power_set (cross_product T_0 T_0)))) /\ (ilf_type (power_set (cross_product T_0 T_0)) (set_type)))) (-. (ilf_type (power_set (cross_product T_0 T_0)) (set_type))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) ### Imply 12 14
% 9.23/9.40 16. (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (-. (ilf_type (power_set (cross_product T_0 T_0)) (set_type))) ### All 15
% 9.23/9.40 17. (-. (empty (power_set (cross_product T_0 T_0)))) (empty (power_set (cross_product T_0 T_0))) ### Axiom
% 9.23/9.40 18. (ilf_type T_0 (set_type)) (-. (ilf_type T_0 (set_type))) ### Axiom
% 9.23/9.40 19. (-. (subset (identity_relation_of T_0) (cross_product T_0 T_0))) (subset (identity_relation_of T_0) (cross_product T_0 T_0)) ### Axiom
% 9.23/9.40 20. ((ilf_type T_0 (set_type)) => (subset (identity_relation_of T_0) (cross_product T_0 T_0))) (-. (subset (identity_relation_of T_0) (cross_product T_0 T_0))) (ilf_type T_0 (set_type)) ### Imply 18 19
% 9.23/9.40 21. (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (ilf_type T_0 (set_type)) (-. (subset (identity_relation_of T_0) (cross_product T_0 T_0))) ### All 20
% 9.23/9.40 22. (-. (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D (cross_product T_0 T_0)))))) (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D (cross_product T_0 T_0))))) ### Axiom
% 9.23/9.40 23. ((subset (identity_relation_of T_0) (cross_product T_0 T_0)) <=> (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D (cross_product T_0 T_0)))))) (-. (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D (cross_product T_0 T_0)))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) ### Equiv 21 22
% 9.23/9.40 24. ((ilf_type (cross_product T_0 T_0) (set_type)) => ((subset (identity_relation_of T_0) (cross_product T_0 T_0)) <=> (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D (cross_product T_0 T_0))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (-. (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D (cross_product T_0 T_0)))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) ### Imply 12 23
% 9.23/9.40 25. (All C, ((ilf_type C (set_type)) => ((subset (identity_relation_of T_0) C) <=> (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D C))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (-. (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D (cross_product T_0 T_0)))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) ### All 24
% 9.23/9.40 26. ((ilf_type (identity_relation_of T_0) (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset (identity_relation_of T_0) C) <=> (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D C)))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (-. (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D (cross_product T_0 T_0)))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, (ilf_type B (set_type))) ### Imply 3 25
% 9.23/9.40 27. (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (-. (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D (cross_product T_0 T_0)))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) ### All 26
% 9.23/9.40 28. (-. (member (identity_relation_of T_0) (power_set (cross_product T_0 T_0)))) (member (identity_relation_of T_0) (power_set (cross_product T_0 T_0))) ### Axiom
% 9.23/9.40 29. ((member (identity_relation_of T_0) (power_set (cross_product T_0 T_0))) <=> (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D (cross_product T_0 T_0)))))) (-. (member (identity_relation_of T_0) (power_set (cross_product T_0 T_0)))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) ### Equiv 27 28
% 9.23/9.40 30. ((ilf_type (cross_product T_0 T_0) (set_type)) => ((member (identity_relation_of T_0) (power_set (cross_product T_0 T_0))) <=> (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D (cross_product T_0 T_0))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (-. (member (identity_relation_of T_0) (power_set (cross_product T_0 T_0)))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) ### Imply 12 29
% 9.23/9.40 31. (All C, ((ilf_type C (set_type)) => ((member (identity_relation_of T_0) (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D C))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (-. (member (identity_relation_of T_0) (power_set (cross_product T_0 T_0)))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) ### All 30
% 9.23/9.42 32. ((ilf_type (identity_relation_of T_0) (set_type)) => (All C, ((ilf_type C (set_type)) => ((member (identity_relation_of T_0) (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D (identity_relation_of T_0)) => (member D C)))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (-. (member (identity_relation_of T_0) (power_set (cross_product T_0 T_0)))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, (ilf_type B (set_type))) ### Imply 3 31
% 9.23/9.42 33. (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (-. (member (identity_relation_of T_0) (power_set (cross_product T_0 T_0)))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) ### All 32
% 9.23/9.42 34. (-. (ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0))))) (ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0)))) ### Axiom
% 9.23/9.42 35. ((ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0)))) <=> (member (identity_relation_of T_0) (power_set (cross_product T_0 T_0)))) (-. (ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) ### Equiv 33 34
% 9.23/9.42 36. (((-. (empty (power_set (cross_product T_0 T_0)))) /\ (ilf_type (power_set (cross_product T_0 T_0)) (set_type))) => ((ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0)))) <=> (member (identity_relation_of T_0) (power_set (cross_product T_0 T_0))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (-. (ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (-. (empty (power_set (cross_product T_0 T_0)))) ### DisjTree 17 16 35
% 9.23/9.42 37. (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type (identity_relation_of T_0) (member_type C)) <=> (member (identity_relation_of T_0) C)))) (-. (empty (power_set (cross_product T_0 T_0)))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (-. (ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) ### All 36
% 9.23/9.42 38. (empty (power_set (cross_product T_0 T_0))) (-. (empty (power_set (cross_product T_0 T_0)))) ### Axiom
% 9.23/9.42 39. ((-. (empty (power_set (cross_product T_0 T_0)))) /\ (ilf_type (power_set (cross_product T_0 T_0)) (set_type))) (empty (power_set (cross_product T_0 T_0))) ### And 38
% 9.23/9.42 40. ((ilf_type (cross_product T_0 T_0) (set_type)) => ((-. (empty (power_set (cross_product T_0 T_0)))) /\ (ilf_type (power_set (cross_product T_0 T_0)) (set_type)))) (empty (power_set (cross_product T_0 T_0))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) ### Imply 12 39
% 9.23/9.42 41. (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (empty (power_set (cross_product T_0 T_0))) ### All 40
% 9.23/9.42 42. ((empty (power_set (cross_product T_0 T_0))) <=> (All C, ((ilf_type C (set_type)) => (-. (member C (power_set (cross_product T_0 T_0))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (-. (ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type (identity_relation_of T_0) (member_type C)) <=> (member (identity_relation_of T_0) C)))) ### Equiv 37 41
% 9.23/9.42 43. ((ilf_type (power_set (cross_product T_0 T_0)) (set_type)) => ((empty (power_set (cross_product T_0 T_0))) <=> (All C, ((ilf_type C (set_type)) => (-. (member C (power_set (cross_product T_0 T_0)))))))) (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type (identity_relation_of T_0) (member_type C)) <=> (member (identity_relation_of T_0) C)))) (-. (ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) ### Imply 16 42
% 9.23/9.43 44. (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (-. (ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0))))) (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type (identity_relation_of T_0) (member_type C)) <=> (member (identity_relation_of T_0) C)))) ### All 43
% 9.23/9.43 45. ((ilf_type (identity_relation_of T_0) (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type (identity_relation_of T_0) (member_type C)) <=> (member (identity_relation_of T_0) C))))) (-. (ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, (ilf_type B (set_type))) ### Imply 3 44
% 9.23/9.43 46. (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (-. (ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0))))) ### All 45
% 9.23/9.43 47. (-. (ilf_type (identity_relation_of T_0) (subset_type (cross_product T_0 T_0)))) (ilf_type (identity_relation_of T_0) (subset_type (cross_product T_0 T_0))) ### Axiom
% 9.23/9.43 48. ((ilf_type (identity_relation_of T_0) (subset_type (cross_product T_0 T_0))) <=> (ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0))))) (-. (ilf_type (identity_relation_of T_0) (subset_type (cross_product T_0 T_0)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) ### Equiv 46 47
% 9.23/9.43 49. ((ilf_type (identity_relation_of T_0) (set_type)) => ((ilf_type (identity_relation_of T_0) (subset_type (cross_product T_0 T_0))) <=> (ilf_type (identity_relation_of T_0) (member_type (power_set (cross_product T_0 T_0)))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (-. (ilf_type (identity_relation_of T_0) (subset_type (cross_product T_0 T_0)))) (All B, (ilf_type B (set_type))) ### Imply 3 48
% 9.23/9.43 50. (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type (cross_product T_0 T_0))) <=> (ilf_type C (member_type (power_set (cross_product T_0 T_0))))))) (All B, (ilf_type B (set_type))) (-. (ilf_type (identity_relation_of T_0) (subset_type (cross_product T_0 T_0)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) ### All 49
% 9.23/9.43 51. ((ilf_type (cross_product T_0 T_0) (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type (cross_product T_0 T_0))) <=> (ilf_type C (member_type (power_set (cross_product T_0 T_0)))))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (-. (ilf_type (identity_relation_of T_0) (subset_type (cross_product T_0 T_0)))) (All B, (ilf_type B (set_type))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) ### Imply 12 50
% 9.23/9.45 52. (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (All B, (ilf_type B (set_type))) (-. (ilf_type (identity_relation_of T_0) (subset_type (cross_product T_0 T_0)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) ### All 51
% 9.23/9.45 53. (-. (ilf_type (identity_relation_of T_0) (relation_type T_0 T_0))) (ilf_type (identity_relation_of T_0) (relation_type T_0 T_0)) ### Axiom
% 9.23/9.45 54. ((ilf_type (identity_relation_of T_0) (subset_type (cross_product T_0 T_0))) => (ilf_type (identity_relation_of T_0) (relation_type T_0 T_0))) (-. (ilf_type (identity_relation_of T_0) (relation_type T_0 T_0))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, (ilf_type B (set_type))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) ### Imply 52 53
% 9.23/9.45 55. (All D, ((ilf_type D (subset_type (cross_product T_0 T_0))) => (ilf_type D (relation_type T_0 T_0)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (ilf_type T_0 (set_type)) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (-. (ilf_type (identity_relation_of T_0) (relation_type T_0 T_0))) ### All 54
% 9.23/9.45 56. ((All D, ((ilf_type D (subset_type (cross_product T_0 T_0))) => (ilf_type D (relation_type T_0 T_0)))) /\ (All E, ((ilf_type E (relation_type T_0 T_0)) => (ilf_type E (subset_type (cross_product T_0 T_0)))))) (-. (ilf_type (identity_relation_of T_0) (relation_type T_0 T_0))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, (ilf_type B (set_type))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) ### And 55
% 9.23/9.45 57. ((ilf_type T_0 (set_type)) => ((All D, ((ilf_type D (subset_type (cross_product T_0 T_0))) => (ilf_type D (relation_type T_0 T_0)))) /\ (All E, ((ilf_type E (relation_type T_0 T_0)) => (ilf_type E (subset_type (cross_product T_0 T_0))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (-. (ilf_type (identity_relation_of T_0) (relation_type T_0 T_0))) (ilf_type T_0 (set_type)) ### Imply 5 56
% 9.23/9.45 58. (All C, ((ilf_type C (set_type)) => ((All D, ((ilf_type D (subset_type (cross_product T_0 C))) => (ilf_type D (relation_type T_0 C)))) /\ (All E, ((ilf_type E (relation_type T_0 C)) => (ilf_type E (subset_type (cross_product T_0 C)))))))) (ilf_type T_0 (set_type)) (-. (ilf_type (identity_relation_of T_0) (relation_type T_0 T_0))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) ### All 57
% 9.23/9.45 59. ((ilf_type T_0 (set_type)) => (All C, ((ilf_type C (set_type)) => ((All D, ((ilf_type D (subset_type (cross_product T_0 C))) => (ilf_type D (relation_type T_0 C)))) /\ (All E, ((ilf_type E (relation_type T_0 C)) => (ilf_type E (subset_type (cross_product T_0 C))))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (-. (ilf_type (identity_relation_of T_0) (relation_type T_0 T_0))) (ilf_type T_0 (set_type)) ### Imply 4 58
% 9.23/9.45 60. (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((All D, ((ilf_type D (subset_type (cross_product B C))) => (ilf_type D (relation_type B C)))) /\ (All E, ((ilf_type E (relation_type B C)) => (ilf_type E (subset_type (cross_product B C)))))))))) (ilf_type T_0 (set_type)) (-. (ilf_type (identity_relation_of T_0) (relation_type T_0 T_0))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) ### All 59
% 9.23/9.45 61. (-. (ilf_type (identity_relation_of T_0) (identity_relation_of_type T_0))) (ilf_type (identity_relation_of T_0) (identity_relation_of_type T_0)) ### Axiom
% 9.23/9.45 62. ((ilf_type (identity_relation_of T_0) (identity_relation_of_type T_0)) <=> (ilf_type (identity_relation_of T_0) (relation_type T_0 T_0))) (-. (ilf_type (identity_relation_of T_0) (identity_relation_of_type T_0))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((All D, ((ilf_type D (subset_type (cross_product B C))) => (ilf_type D (relation_type B C)))) /\ (All E, ((ilf_type E (relation_type B C)) => (ilf_type E (subset_type (cross_product B C)))))))))) ### Equiv 60 61
% 9.23/9.45 63. ((ilf_type (identity_relation_of T_0) (set_type)) => ((ilf_type (identity_relation_of T_0) (identity_relation_of_type T_0)) <=> (ilf_type (identity_relation_of T_0) (relation_type T_0 T_0)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((All D, ((ilf_type D (subset_type (cross_product B C))) => (ilf_type D (relation_type B C)))) /\ (All E, ((ilf_type E (relation_type B C)) => (ilf_type E (subset_type (cross_product B C)))))))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) (-. (ilf_type (identity_relation_of T_0) (identity_relation_of_type T_0))) (All B, (ilf_type B (set_type))) ### Imply 3 62
% 9.23/9.45 64. (All C, ((ilf_type C (set_type)) => ((ilf_type C (identity_relation_of_type T_0)) <=> (ilf_type C (relation_type T_0 T_0))))) (All B, (ilf_type B (set_type))) (-. (ilf_type (identity_relation_of T_0) (identity_relation_of_type T_0))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (ilf_type T_0 (set_type)) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((All D, ((ilf_type D (subset_type (cross_product B C))) => (ilf_type D (relation_type B C)))) /\ (All E, ((ilf_type E (relation_type B C)) => (ilf_type E (subset_type (cross_product B C)))))))))) ### All 63
% 9.23/9.46 65. ((ilf_type T_0 (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (identity_relation_of_type T_0)) <=> (ilf_type C (relation_type T_0 T_0)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((All D, ((ilf_type D (subset_type (cross_product B C))) => (ilf_type D (relation_type B C)))) /\ (All E, ((ilf_type E (relation_type B C)) => (ilf_type E (subset_type (cross_product B C)))))))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) (-. (ilf_type (identity_relation_of T_0) (identity_relation_of_type T_0))) (All B, (ilf_type B (set_type))) (ilf_type T_0 (set_type)) ### Imply 1 64
% 9.23/9.46 66. (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (identity_relation_of_type B)) <=> (ilf_type C (relation_type B B))))))) (ilf_type T_0 (set_type)) (All B, (ilf_type B (set_type))) (-. (ilf_type (identity_relation_of T_0) (identity_relation_of_type T_0))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((All D, ((ilf_type D (subset_type (cross_product B C))) => (ilf_type D (relation_type B C)))) /\ (All E, ((ilf_type E (relation_type B C)) => (ilf_type E (subset_type (cross_product B C)))))))))) ### All 65
% 9.23/9.46 67. (-. ((ilf_type T_0 (set_type)) => (ilf_type (identity_relation_of T_0) (identity_relation_of_type T_0)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((All D, ((ilf_type D (subset_type (cross_product B C))) => (ilf_type D (relation_type B C)))) /\ (All E, ((ilf_type E (relation_type B C)) => (ilf_type E (subset_type (cross_product B C)))))))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (identity_relation_of_type B)) <=> (ilf_type C (relation_type B B))))))) ### NotImply 66
% 9.23/9.46 68. (-. (All B, ((ilf_type B (set_type)) => (ilf_type (identity_relation_of B) (identity_relation_of_type B))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (identity_relation_of_type B)) <=> (ilf_type C (relation_type B B))))))) (All B, (ilf_type B (set_type))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((ilf_type C (subset_type B)) <=> (ilf_type C (member_type (power_set B)))))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => (ilf_type (cross_product B C) (set_type)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((subset B C) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => (subset (identity_relation_of B) (cross_product B B)))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((member B (power_set C)) <=> (All D, ((ilf_type D (set_type)) => ((member D B) => (member D C))))))))) (All B, ((ilf_type B (set_type)) => ((-. (empty (power_set B))) /\ (ilf_type (power_set B) (set_type))))) (All B, ((ilf_type B (set_type)) => ((empty B) <=> (All C, ((ilf_type C (set_type)) => (-. (member C B))))))) (All B, ((ilf_type B (set_type)) => (All C, (((-. (empty C)) /\ (ilf_type C (set_type))) => ((ilf_type B (member_type C)) <=> (member B C)))))) (All B, ((ilf_type B (set_type)) => (All C, ((ilf_type C (set_type)) => ((All D, ((ilf_type D (subset_type (cross_product B C))) => (ilf_type D (relation_type B C)))) /\ (All E, ((ilf_type E (relation_type B C)) => (ilf_type E (subset_type (cross_product B C)))))))))) ### NotAllEx 67
% 9.23/9.46 % SZS output end Proof
% 9.23/9.46 (* END-PROOF *)
%------------------------------------------------------------------------------