Theorem eluni2f

Description: Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	eluni2f.1	⊢ Ⅎ 𝑥 𝐴
	eluni2f.2	⊢ Ⅎ 𝑥 𝐵
Assertion	eluni2f	⊢ ( 𝐴 ∈ ∪ 𝐵 ↔ ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		eluni2f.1	⊢ Ⅎ 𝑥 𝐴
2		eluni2f.2	⊢ Ⅎ 𝑥 𝐵
3		exancom	⊢ ( ∃ 𝑥 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥 ) )
4	1 2	elunif	⊢ ( 𝐴 ∈ ∪ 𝐵 ↔ ∃ 𝑥 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) )
5		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥 ) )
6	3 4 5	3bitr4i	⊢ ( 𝐴 ∈ ∪ 𝐵 ↔ ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 )