Theorem unfid

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

	Ref	Expression
Hypotheses	unfid.1	$$⊢\phi \to A\in \mathrm{Fin}$$
	unfid.2	$$⊢\phi \to B\in \mathrm{Fin}$$
Assertion	unfid	$$⊢\phi \to A\cup B\in \mathrm{Fin}$$

Proof

Step	Hyp	Ref	Expression
1		unfid.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		unfid.2	$$⊢\phi \to B\in \mathrm{Fin}$$
3		unfi	$$⊢\left(A\in \mathrm{Fin}\wedge B\in \mathrm{Fin}\right)\to A\cup B\in \mathrm{Fin}$$
4	1 2 3	syl2anc	$$⊢\phi \to A\cup B\in \mathrm{Fin}$$