Theorem bj-elsn12g 34377

Description: Join of elsng 4574 and elsn2g 4596. (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 4574	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2		elsn2g 4596	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	jaoi 853	1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∨ wo 843   = wceq 1536   ∈ wcel 2113  {csn 4560

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-sn 4561

This theorem is referenced by: (None)