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Theorem bj-elid3 33946
 Description: Characterization of the couples in 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 33944 . 2 (⟨𝐴, 𝐵⟩ ∈ I ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)))
2 opelxp 5443 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
32anbi1i 614 . . 3 ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)))
4 op1stg 7513 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
5 op2ndg 7514 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
64, 5eqeq12d 2793 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩) ↔ 𝐴 = 𝐵))
76pm5.32i 567 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
8 simpl 475 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
98anim1i 605 . . . . 5 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐴 = 𝐵))
10 simpl 475 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
11 eleq1 2853 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
1211biimpac 471 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
13 simpr 477 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
1410, 12, 13jca31 507 . . . . 5 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
159, 14impbii 201 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
167, 15bitri 267 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
173, 16bitri 267 . 2 ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
181, 17bitri 267 1 (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  Vcvv 3415  ⟨cop 4447   I cid 5311   × cxp 5405  ‘cfv 6188  1st c1st 7499  2nd c2nd 7500 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-iota 6152  df-fun 6190  df-fv 6196  df-1st 7501  df-2nd 7502 This theorem is referenced by:  bj-elid4  33947  bj-eldiag2  33952
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