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Theorem cnvdif 6165
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 6124 . 2 Rel (𝐴𝐵)
2 difss 4145 . . 3 (𝐴𝐵) ⊆ 𝐴
3 relcnv 6124 . . 3 Rel 𝐴
4 relss 5793 . . 3 ((𝐴𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴𝐵)))
52, 3, 4mp2 9 . 2 Rel (𝐴𝐵)
6 eldif 3972 . . 3 (⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
7 vex 3481 . . . 4 𝑥 ∈ V
8 vex 3481 . . . 4 𝑦 ∈ V
97, 8opelcnv 5894 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵))
10 eldif 3972 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
117, 8opelcnv 5894 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
127, 8opelcnv 5894 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1312notbii 320 . . . . 5 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1411, 13anbi12i 628 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
1510, 14bitri 275 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
166, 9, 153bitr4i 303 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵))
171, 5, 16eqrelriiv 5802 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1536  wcel 2105  cdif 3959  wss 3962  cop 4636  ccnv 5687  Rel wrel 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696
This theorem is referenced by:  cnvin  6166  gtiso  32715  gsumhashmul  33046  mthmpps  35566  cnvnonrel  43577
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