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Theorem cnvdif 5980
Description: Distributive law for converse over class 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 5945 . 2 Rel (𝐴𝐵)
2 difss 4083 . . 3 (𝐴𝐵) ⊆ 𝐴
3 relcnv 5945 . . 3 Rel 𝐴
4 relss 5633 . . 3 ((𝐴𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴𝐵)))
52, 3, 4mp2 9 . 2 Rel (𝐴𝐵)
6 eldif 3918 . . 3 (⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
7 vex 3472 . . . 4 𝑥 ∈ V
8 vex 3472 . . . 4 𝑦 ∈ V
97, 8opelcnv 5729 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵))
10 eldif 3918 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
117, 8opelcnv 5729 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
127, 8opelcnv 5729 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1312notbii 323 . . . . 5 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1411, 13anbi12i 629 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
1510, 14bitri 278 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
166, 9, 153bitr4i 306 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵))
171, 5, 16eqrelriiv 5640 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1538  wcel 2114  cdif 3905  wss 3908  cop 4545  ccnv 5531  Rel wrel 5537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540
This theorem is referenced by:  cnvin  5981  gtiso  30444  mthmpps  32903  cnvnonrel  40222
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