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Theorem opeldifid 30474
 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 5662 . . . 4 (Rel 𝐴 → Rel (𝐴 ∖ I ))
2 brrelex2 5580 . . . 4 ((Rel (𝐴 ∖ I ) ∧ 𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V)
31, 2sylan 583 . . 3 ((Rel 𝐴𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V)
4 brrelex2 5580 . . . 4 ((Rel 𝐴𝑋𝐴𝑌) → 𝑌 ∈ V)
54adantrr 716 . . 3 ((Rel 𝐴 ∧ (𝑋𝐴𝑌𝑋𝑌)) → 𝑌 ∈ V)
6 brdif 5089 . . . 4 (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌))
7 ideqg 5697 . . . . . 6 (𝑌 ∈ V → (𝑋 I 𝑌𝑋 = 𝑌))
87necon3bbid 2988 . . . . 5 (𝑌 ∈ V → (¬ 𝑋 I 𝑌𝑋𝑌))
98anbi2d 631 . . . 4 (𝑌 ∈ V → ((𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌) ↔ (𝑋𝐴𝑌𝑋𝑌)))
106, 9syl5bb 286 . . 3 (𝑌 ∈ V → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌𝑋𝑌)))
113, 5, 10pm5.21nd 801 . 2 (Rel 𝐴 → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌𝑋𝑌)))
12 df-br 5037 . 2 (𝑋(𝐴 ∖ I )𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ))
13 df-br 5037 . . 3 (𝑋𝐴𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝐴)
1413anbi1i 626 . 2 ((𝑋𝐴𝑌𝑋𝑌) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴𝑋𝑌))
1511, 12, 143bitr3g 316 1 (Rel 𝐴 → (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴𝑋𝑌)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111   ≠ wne 2951  Vcvv 3409   ∖ cdif 3857  ⟨cop 4531   class class class wbr 5036   I cid 5433  Rel wrel 5533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535 This theorem is referenced by:  qtophaus  31320
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