Theorem bj-elsnb 37105

Description: Biconditional version of elsng 4587. (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 3457	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 ∈ V)
2		elsng 4587	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	biadanii 821	1 ⊢ (𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  {csn 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-sn 4574

This theorem is referenced by: (None)