Theorem elunsn 4650

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 4119	. 2 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐶}))
2		elsng 4606	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐶} ↔ 𝐴 = 𝐶))
3	2	orbi2d 915	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐶)))
4	1, 3	bitrid 283	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ∪ cun 3915  {csn 4592

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-un 3922  df-sn 4593

This theorem is referenced by:  f1ounsn  7250  chnub  32945  cycpmco2lem7  33096