Theorem prct

Description: An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016)

	Ref	Expression
Assertion	prct	$$⊢\left(A\in V\wedge B\in W\right)\to \left\{A,B\right\}\preccurlyeq \mathrm{\omega }$$

Proof

Step	Hyp	Ref	Expression
1		df-pr	$$⊢\left\{A,B\right\}=\left\{A\right\}\cup \left\{B\right\}$$
2		snct	$$⊢A\in V\to \left\{A\right\}\preccurlyeq \mathrm{\omega }$$
3		snct	$$⊢B\in W\to \left\{B\right\}\preccurlyeq \mathrm{\omega }$$
4		unctb	$$⊢\left(\left\{A\right\}\preccurlyeq \mathrm{\omega }\wedge \left\{B\right\}\preccurlyeq \mathrm{\omega }\right)\to \left(\left\{A\right\}\cup \left\{B\right\}\right)\preccurlyeq \mathrm{\omega }$$
5	2 3 4	syl2an	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left\{A\right\}\cup \left\{B\right\}\right)\preccurlyeq \mathrm{\omega }$$
6	1 5	eqbrtrid	$$⊢\left(A\in V\wedge B\in W\right)\to \left\{A,B\right\}\preccurlyeq \mathrm{\omega }$$