Theorem bnj1405

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj1405.1	⊢ ( 𝜑 → 𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 )
Assertion	bnj1405	⊢ ( 𝜑 → ∃ 𝑦 ∈ 𝐴 𝑋 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		bnj1405.1	⊢ ( 𝜑 → 𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 )
2		eliun	⊢ ( 𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ↔ ∃ 𝑦 ∈ 𝐴 𝑋 ∈ 𝐵 )
3	1 2	sylib	⊢ ( 𝜑 → ∃ 𝑦 ∈ 𝐴 𝑋 ∈ 𝐵 )