Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opeldifid Structured version   Visualization version   GIF version

Theorem opeldifid 32265
Description: Ordered pair elementhood outside of the diagonal. (Contributed by Thierry Arnoux, 1-Jan-2020.)
Assertion
Ref Expression
opeldifid (Rel 𝐴 → (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴𝑋𝑌)))

Proof of Theorem opeldifid
StepHypRef Expression
1 reldif 5805 . . . 4 (Rel 𝐴 → Rel (𝐴 ∖ I ))
2 brrelex2 5720 . . . 4 ((Rel (𝐴 ∖ I ) ∧ 𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V)
31, 2sylan 579 . . 3 ((Rel 𝐴𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V)
4 brrelex2 5720 . . . 4 ((Rel 𝐴𝑋𝐴𝑌) → 𝑌 ∈ V)
54adantrr 714 . . 3 ((Rel 𝐴 ∧ (𝑋𝐴𝑌𝑋𝑌)) → 𝑌 ∈ V)
6 brdif 5191 . . . 4 (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌))
7 ideqg 5841 . . . . . 6 (𝑌 ∈ V → (𝑋 I 𝑌𝑋 = 𝑌))
87necon3bbid 2970 . . . . 5 (𝑌 ∈ V → (¬ 𝑋 I 𝑌𝑋𝑌))
98anbi2d 628 . . . 4 (𝑌 ∈ V → ((𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌) ↔ (𝑋𝐴𝑌𝑋𝑌)))
106, 9bitrid 283 . . 3 (𝑌 ∈ V → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌𝑋𝑌)))
113, 5, 10pm5.21nd 799 . 2 (Rel 𝐴 → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌𝑋𝑌)))
12 df-br 5139 . 2 (𝑋(𝐴 ∖ I )𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ))
13 df-br 5139 . . 3 (𝑋𝐴𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝐴)
1413anbi1i 623 . 2 ((𝑋𝐴𝑌𝑋𝑌) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴𝑋𝑌))
1511, 12, 143bitr3g 313 1 (Rel 𝐴 → (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴𝑋𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wcel 2098  wne 2932  Vcvv 3466  cdif 3937  cop 4626   class class class wbr 5138   I cid 5563  Rel wrel 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673
This theorem is referenced by:  qtophaus  33271
  Copyright terms: Public domain W3C validator