Theorem bj-elsn12g 37083

Description: Join of elsng 4620 and elsn2g 4645. (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 4620	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2		elsn2g 4645	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	jaoi 857	1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1540   ∈ wcel 2109  {csn 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-sn 4607

This theorem is referenced by: (None)