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Theorem cnvdif 4757
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 4730 . 2 Rel (𝐴𝐵)
2 difss 3097 . . 3 (𝐴𝐵) ⊆ 𝐴
3 relcnv 4730 . . 3 Rel 𝐴
4 relss 4454 . . 3 ((𝐴𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴𝐵)))
52, 3, 4mp2 16 . 2 Rel (𝐴𝐵)
6 eldif 2954 . . 3 (⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
7 vex 2577 . . . 4 𝑥 ∈ V
8 vex 2577 . . . 4 𝑦 ∈ V
97, 8opelcnv 4544 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵))
10 eldif 2954 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
117, 8opelcnv 4544 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
127, 8opelcnv 4544 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1312notbii 604 . . . . 5 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1411, 13anbi12i 441 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
1510, 14bitri 177 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
166, 9, 153bitr4i 205 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵))
171, 5, 16eqrelriiv 4461 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101   = wceq 1259  wcel 1409  cdif 2941  wss 2944  cop 3405  ccnv 4371  Rel wrel 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380
This theorem is referenced by: (None)
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