Theorem bj-elsn12g 37975

Description: Join of elsng 4597 and elsn2g 4624. (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 4597	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2		elsn2g 4624	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	jaoi 871	1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1569   ∈ wcel 2144  {csn 4583

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-sn 4584

This theorem is used by: (None)