Theorem 0iun

Description: An empty indexed union is empty. (Contributed by NM, 4-Dec-2004) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	0iun	⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Proof

Step	Hyp	Ref	Expression
1		rex0	⊢ ¬ ∃ 𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃ 𝑥 ∈ ∅ 𝑦 ∈ 𝐴 )
3	1 2	mtbir	⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4	3	nel0	⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅