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Theorem bj-elid4 34897
 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 7738 . 2 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 eleq1 2839 . . . 4 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
32adantl 485 . . 3 ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩) → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
4 fvex 6676 . . . . 5 (2nd𝐴) ∈ V
54inex2 5192 . . . 4 ((1st𝐴) ∩ (2nd𝐴)) ∈ V
6 bj-opelid 34885 . . . 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 282 . 2 ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))
91, 8mpdan 686 1 (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∩ cin 3859  ⟨cop 4531   I cid 5433   × cxp 5526  ‘cfv 6340  1st c1st 7697  2nd c2nd 7698 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-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 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-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  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-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6299  df-fun 6342  df-fv 6348  df-1st 7699  df-2nd 7700 This theorem is referenced by:  bj-elid5  34898  bj-elid6  34899  bj-eldiag  34905
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