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Theorem opeldifid 29298
 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 5209 . . . 4 (Rel 𝐴 → Rel (𝐴 ∖ I ))
2 brrelex2 5127 . . . 4 ((Rel (𝐴 ∖ I ) ∧ 𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V)
31, 2sylan 488 . . 3 ((Rel 𝐴𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V)
4 brrelex2 5127 . . . 4 ((Rel 𝐴𝑋𝐴𝑌) → 𝑌 ∈ V)
54adantrr 752 . . 3 ((Rel 𝐴 ∧ (𝑋𝐴𝑌𝑋𝑌)) → 𝑌 ∈ V)
6 brdif 4675 . . . 4 (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌))
7 ideqg 5243 . . . . . 6 (𝑌 ∈ V → (𝑋 I 𝑌𝑋 = 𝑌))
87necon3bbid 2827 . . . . 5 (𝑌 ∈ V → (¬ 𝑋 I 𝑌𝑋𝑌))
98anbi2d 739 . . . 4 (𝑌 ∈ V → ((𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌) ↔ (𝑋𝐴𝑌𝑋𝑌)))
106, 9syl5bb 272 . . 3 (𝑌 ∈ V → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌𝑋𝑌)))
113, 5, 10pm5.21nd 940 . 2 (Rel 𝐴 → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌𝑋𝑌)))
12 df-br 4624 . 2 (𝑋(𝐴 ∖ I )𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ))
13 df-br 4624 . . 3 (𝑋𝐴𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝐴)
1413anbi1i 730 . 2 ((𝑋𝐴𝑌𝑋𝑌) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴𝑋𝑌))
1511, 12, 143bitr3g 302 1 (Rel 𝐴 → (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴𝑋𝑌)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∈ wcel 1987   ≠ wne 2790  Vcvv 3190   ∖ cdif 3557  ⟨cop 4161   class class class wbr 4623   I cid 4994  Rel wrel 5089 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091 This theorem is referenced by:  qtophaus  29727
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