ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnvdif GIF version

Theorem cnvdif 4881
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 4853 . 2 Rel (𝐴𝐵)
2 difss 3149 . . 3 (𝐴𝐵) ⊆ 𝐴
3 relcnv 4853 . . 3 Rel 𝐴
4 relss 4564 . . 3 ((𝐴𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴𝐵)))
52, 3, 4mp2 16 . 2 Rel (𝐴𝐵)
6 eldif 3030 . . 3 (⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
7 vex 2644 . . . 4 𝑥 ∈ V
8 vex 2644 . . . 4 𝑦 ∈ V
97, 8opelcnv 4659 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵))
10 eldif 3030 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
117, 8opelcnv 4659 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
127, 8opelcnv 4659 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1312notbii 635 . . . . 5 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1411, 13anbi12i 451 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
1510, 14bitri 183 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
166, 9, 153bitr4i 211 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵))
171, 5, 16eqrelriiv 4571 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1299  wcel 1448  cdif 3018  wss 3021  cop 3477  ccnv 4476  Rel wrel 4482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-rel 4484  df-cnv 4485
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator