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Theorem cnvdif 6094
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 6054 . 2 Rel (𝐴𝐵)
2 difss 4089 . . 3 (𝐴𝐵) ⊆ 𝐴
3 relcnv 6054 . . 3 Rel 𝐴
4 relss 5735 . . 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 3447 . . . 4 𝑥 ∈ V
8 vex 3447 . . . 4 𝑦 ∈ V
97, 8opelcnv 5835 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵))
10 eldif 3918 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
117, 8opelcnv 5835 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
127, 8opelcnv 5835 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1312notbii 319 . . . . 5 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1411, 13anbi12i 627 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
1510, 14bitri 274 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
166, 9, 153bitr4i 302 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵))
171, 5, 16eqrelriiv 5744 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1541  wcel 2106  cdif 3905  wss 3908  cop 4590  ccnv 5630  Rel wrel 5636
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-rel 5638  df-cnv 5639
This theorem is referenced by:  cnvin  6095  gtiso  31498  gsumhashmul  31781  mthmpps  34045  cnvnonrel  41802
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