Theorem elunsn 4654

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 4115	. 2 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐶}))
2		elsng 4608	. . 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)))

Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ∪ cun 3911  {csn 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-sn 4595

This theorem is referenced by:  f1ounsn  7271  chnub  18678  nn0mnfxrd  33037  cycpmco2lem7  33393