Theorem eliund

Description: Membership in indexed union. (Contributed by Glauco Siliprandi, 15-Feb-2025)

	Ref	Expression
Hypothesis	eliund.1	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 )
Assertion	eliund	⊢ ( 𝜑 → 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eliund.1	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 )
2		eliun	⊢ ( 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 )
3	1 2	sylibr	⊢ ( 𝜑 → 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 )