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Theorem bj-elid3 33390
Description: Characterization of the elements of I. (Contributed by BJ, 29-Mar-2020.)
Assertion
Ref Expression
bj-elid3 (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Proof of Theorem bj-elid3
StepHypRef Expression
1 bj-elid 33388 . 2 (⟨𝐴, 𝐵⟩ ∈ I ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)))
2 opelxp 5295 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
32anbi1i 733 . . 3 ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)))
4 op1stg 7337 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
5 op2ndg 7338 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
64, 5eqeq12d 2767 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩) ↔ 𝐴 = 𝐵))
76pm5.32i 672 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
8 simpl 474 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
98anim1i 593 . . . . 5 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐴 = 𝐵))
10 simpl 474 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
11 eleq1 2819 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
1211biimpac 504 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
13 simpr 479 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
1410, 12, 13jca31 558 . . . . 5 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
159, 14impbii 199 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
167, 15bitri 264 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
173, 16bitri 264 . 2 ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
181, 17bitri 264 1 (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1624  wcel 2131  Vcvv 3332  cop 4319   I cid 5165   × cxp 5256  cfv 6041  1st c1st 7323  2nd c2nd 7324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fv 6049  df-1st 7325  df-2nd 7326
This theorem is referenced by:  bj-eldiag2  33395
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