Theorem unifi3

Description: If a union is finite, then all its elements are finite. See unifi . (Contributed by Thierry Arnoux, 27-Aug-2017)

	Ref	Expression
Assertion	unifi3	⊢ ( ∪ 𝐴 ∈ Fin → 𝐴 ⊆ Fin )

Proof

Step	Hyp	Ref	Expression
1		elssuni	⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴 )
2		ssfi	⊢ ( ( ∪ 𝐴 ∈ Fin ∧ 𝑥 ⊆ ∪ 𝐴 ) → 𝑥 ∈ Fin )
3	2	ex	⊢ ( ∪ 𝐴 ∈ Fin → ( 𝑥 ⊆ ∪ 𝐴 → 𝑥 ∈ Fin ) )
4	1 3	syl5	⊢ ( ∪ 𝐴 ∈ Fin → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ Fin ) )
5	4	ssrdv	⊢ ( ∪ 𝐴 ∈ Fin → 𝐴 ⊆ Fin )