Theorem prfi

Description: An unordered pair is finite. (Contributed by NM, 22-Aug-2008)

	Ref	Expression
Assertion	prfi	$$⊢\left\{A,B\right\}\in \mathrm{Fin}$$

Proof

Step	Hyp	Ref	Expression
1		df-pr	$$⊢\left\{A,B\right\}=\left\{A\right\}\cup \left\{B\right\}$$
2		snfi	$$⊢\left\{A\right\}\in \mathrm{Fin}$$
3		snfi	$$⊢\left\{B\right\}\in \mathrm{Fin}$$
4		unfi	$$⊢\left(\left\{A\right\}\in \mathrm{Fin}\wedge \left\{B\right\}\in \mathrm{Fin}\right)\to \left\{A\right\}\cup \left\{B\right\}\in \mathrm{Fin}$$
5	2 3 4	mp2an	$$⊢\left\{A\right\}\cup \left\{B\right\}\in \mathrm{Fin}$$
6	1 5	eqeltri	$$⊢\left\{A,B\right\}\in \mathrm{Fin}$$