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Theorem opeldifid 29740
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 5394 . . . 4 (Rel 𝐴 → Rel (𝐴 ∖ I ))
2 brrelex2 5314 . . . 4 ((Rel (𝐴 ∖ I ) ∧ 𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V)
31, 2sylan 489 . . 3 ((Rel 𝐴𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V)
4 brrelex2 5314 . . . 4 ((Rel 𝐴𝑋𝐴𝑌) → 𝑌 ∈ V)
54adantrr 755 . . 3 ((Rel 𝐴 ∧ (𝑋𝐴𝑌𝑋𝑌)) → 𝑌 ∈ V)
6 brdif 4857 . . . 4 (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌))
7 ideqg 5429 . . . . . 6 (𝑌 ∈ V → (𝑋 I 𝑌𝑋 = 𝑌))
87necon3bbid 2969 . . . . 5 (𝑌 ∈ V → (¬ 𝑋 I 𝑌𝑋𝑌))
98anbi2d 742 . . . 4 (𝑌 ∈ V → ((𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌) ↔ (𝑋𝐴𝑌𝑋𝑌)))
106, 9syl5bb 272 . . 3 (𝑌 ∈ V → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌𝑋𝑌)))
113, 5, 10pm5.21nd 979 . 2 (Rel 𝐴 → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌𝑋𝑌)))
12 df-br 4805 . 2 (𝑋(𝐴 ∖ I )𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ))
13 df-br 4805 . . 3 (𝑋𝐴𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝐴)
1413anbi1i 733 . 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 383  wcel 2139  wne 2932  Vcvv 3340  cdif 3712  cop 4327   class class class wbr 4804   I cid 5173  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273
This theorem is referenced by:  qtophaus  30233
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