Theorem unfid

Description: The union of two finite sets is finite. (Contributed by Glauco Siliprandi, 5-Feb-2022)

	Ref	Expression
Hypotheses	unfid.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	unfid.2	⊢ ( 𝜑 → 𝐵 ∈ Fin )
Assertion	unfid	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		unfid.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		unfid.2	⊢ ( 𝜑 → 𝐵 ∈ Fin )
3		unfi	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( 𝐴 ∪ 𝐵 ) ∈ Fin )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) ∈ Fin )