Theorem bj-opelidb1 36524

Description: Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 36523 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 643	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ∧ 𝐵 ∈ V))
2		bj-opelidb 36523	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
3		eleq1 2813	. . . 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 303	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466  ⟨cop 4626   I cid 5563

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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-opab 5201  df-id 5564

This theorem is referenced by:  bj-idres  36531  bj-elid3  36538  bj-eldiag2  36548