Theorem bj-elsn12g 35936

Description: Join of elsng 4642 and elsn2g 4666. (Contributed by BJ, 18-Nov-2023.)

Assertion

Ref	Expression
bj-elsn12g	⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem bj-elsn12g

Step	Hyp	Ref	Expression
1		elsng 4642	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2		elsn2g 4666	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	jaoi 855	1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 845   = wceq 1541   ∈ wcel 2106  {csn 4628

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 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-sn 4629

This theorem is referenced by: (None)