Theorem bj-elsnb 35521

Description: Biconditional version of elsng 4599. (Contributed by BJ, 18-Nov-2023.)

Assertion

Ref	Expression
bj-elsnb	⊢ (𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Proof of Theorem bj-elsnb

Step	Hyp	Ref	Expression
1		elex 3462	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 ∈ V)
2		elsng 4599	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	biadanii 820	1 ⊢ (𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3444  {csn 4585

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3446  df-sn 4586

This theorem is referenced by: (None)