Theorem bj-opelidb1 37465

Description: Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 37464 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 647	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ∧ 𝐵 ∈ V))
2		bj-opelidb 37464	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
3		eleq1 2825	. . . 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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   I cid 5516

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-opab 5149  df-id 5517

This theorem is referenced by:  bj-idres  37472  bj-elid3  37479  bj-eldiag2  37489