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Theorem cnvdif 4653
Description: Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
cnvdif (AB) = (AB)

Proof of Theorem cnvdif
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 4626 . 2 Rel (AB)
2 difss 3043 . . 3 (AB) ⊆ A
3 relcnv 4626 . . 3 Rel A
4 relss 4350 . . 3 ((AB) ⊆ A → (Rel A → Rel (AB)))
52, 3, 4mp2 16 . 2 Rel (AB)
6 eldif 2900 . . 3 (⟨y, x (AB) ↔ (⟨y, x A ¬ ⟨y, x B))
7 vex 2534 . . . 4 x V
8 vex 2534 . . . 4 y V
97, 8opelcnv 4440 . . 3 (⟨x, y (AB) ↔ ⟨y, x (AB))
10 eldif 2900 . . . 4 (⟨x, y (AB) ↔ (⟨x, y A ¬ ⟨x, y B))
117, 8opelcnv 4440 . . . . 5 (⟨x, y A ↔ ⟨y, x A)
127, 8opelcnv 4440 . . . . . 6 (⟨x, y B ↔ ⟨y, x B)
1312notbii 581 . . . . 5 (¬ ⟨x, y B ↔ ¬ ⟨y, x B)
1411, 13anbi12i 436 . . . 4 ((⟨x, y A ¬ ⟨x, y B) ↔ (⟨y, x A ¬ ⟨y, x B))
1510, 14bitri 173 . . 3 (⟨x, y (AB) ↔ (⟨y, x A ¬ ⟨y, x B))
166, 9, 153bitr4i 201 . 2 (⟨x, y (AB) ↔ ⟨x, y (AB))
171, 5, 16eqrelriiv 4357 1 (AB) = (AB)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1226   wcel 1370  cdif 2887  wss 2890  cop 3349  ccnv 4267  Rel wrel 4273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-cnv 4276
This theorem is referenced by: (None)
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