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Theorem bj-elid4 36704
Description: Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-elid4 (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))

Proof of Theorem bj-elid4
StepHypRef Expression
1 1st2nd2 8030 . 2 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 eleq1 2813 . . . 4 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
32adantl 480 . . 3 ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩) → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
4 fvex 6905 . . . . 5 (2nd𝐴) ∈ V
54inex2 5313 . . . 4 ((1st𝐴) ∩ (2nd𝐴)) ∈ V
6 bj-opelid 36692 . . . 4 (((1st𝐴) ∩ (2nd𝐴)) ∈ V → (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ↔ (1st𝐴) = (2nd𝐴)))
75, 6mp1i 13 . . 3 ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩) → (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ↔ (1st𝐴) = (2nd𝐴)))
83, 7bitrd 278 . 2 ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))
91, 8mpdan 685 1 (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  Vcvv 3463  cin 3938  cop 4630   I cid 5569   × cxp 5670  cfv 6543  1st c1st 7989  2nd c2nd 7990
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7991  df-2nd 7992
This theorem is referenced by:  bj-elid5  36705  bj-elid6  36706  bj-eldiag  36712
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