Theorem elunsn 4649

Description: Elementhood in a union with a singleton. (Contributed by Thierry Arnoux, 14-Dec-2023.)

Assertion

Ref	Expression
elunsn	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐶)))

Proof of Theorem elunsn

Step	Hyp	Ref	Expression
1		elun 4107	. 2 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐶}))
2		elsng 4603	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐶} ↔ 𝐴 = 𝐶))
3	2	orbi2d 928	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐶)))
4	1, 3	bitrid 286	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ∪ cun 3903  {csn 4589

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3910  df-sn 4590

This theorem is used by:  f1ounsn  7270  chnub  18682  nn0mnfxrd  33105  cycpmco2lem7  33461