Theorem bj-opelidb1 35251

Description: Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 35250 where only the sethood of the first component is expressed. (Contributed by BJ, 27-Dec-2023.)

Assertion

Ref	Expression
bj-opelidb1	⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Proof of Theorem bj-opelidb1

Step	Hyp	Ref	Expression
1		an32 642	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ∧ 𝐵 ∈ V))
2		bj-opelidb 35250	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
3		eleq1 2826	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
4	3	biimpac 478	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
5	4	pm4.71i 559	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ↔ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ∧ 𝐵 ∈ V))
6	1, 2, 5	3bitr4i 302	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564   I cid 5479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-id 5480

This theorem is referenced by:  bj-idres  35258  bj-elid3  35265  bj-eldiag2  35275