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Theorem opeldifid 29538
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 5271 . . . 4 (Rel 𝐴 → Rel (𝐴 ∖ I ))
2 brrelex2 5191 . . . 4 ((Rel (𝐴 ∖ I ) ∧ 𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V)
31, 2sylan 487 . . 3 ((Rel 𝐴𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V)
4 brrelex2 5191 . . . 4 ((Rel 𝐴𝑋𝐴𝑌) → 𝑌 ∈ V)
54adantrr 753 . . 3 ((Rel 𝐴 ∧ (𝑋𝐴𝑌𝑋𝑌)) → 𝑌 ∈ V)
6 brdif 4738 . . . 4 (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌))
7 ideqg 5306 . . . . . 6 (𝑌 ∈ V → (𝑋 I 𝑌𝑋 = 𝑌))
87necon3bbid 2860 . . . . 5 (𝑌 ∈ V → (¬ 𝑋 I 𝑌𝑋𝑌))
98anbi2d 740 . . . 4 (𝑌 ∈ V → ((𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌) ↔ (𝑋𝐴𝑌𝑋𝑌)))
106, 9syl5bb 272 . . 3 (𝑌 ∈ V → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌𝑋𝑌)))
113, 5, 10pm5.21nd 961 . 2 (Rel 𝐴 → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌𝑋𝑌)))
12 df-br 4686 . 2 (𝑋(𝐴 ∖ I )𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ))
13 df-br 4686 . . 3 (𝑋𝐴𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝐴)
1413anbi1i 731 . 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 2030  wne 2823  Vcvv 3231  cdif 3604  cop 4216   class class class wbr 4685   I cid 5052  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150
This theorem is referenced by:  qtophaus  30031
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