Theorem eluni2

Description: Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999)

	Ref	Expression
Assertion	eluni2	⊢ ( 𝐴 ∈ ∪ 𝐵 ↔ ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		exancom	⊢ ( ∃ 𝑥 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥 ) )
2		eluni	⊢ ( 𝐴 ∈ ∪ 𝐵 ↔ ∃ 𝑥 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) )
3		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥 ) )
4	1 2 3	3bitr4i	⊢ ( 𝐴 ∈ ∪ 𝐵 ↔ ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 )