Theorem elunsn 4651

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 4605	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐶} ↔ 𝐴 = 𝐶))
3	2	orbi2d 929	. 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 861   = wceq 1570   ∈ wcel 2146   ∪ cun 3904  {csn 4591

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-sn 4592

This theorem is used by:  f1ounsn  7279  chnub  18704  nn0mnfxrd  33170  cycpmco2lem7  33520