Theorem xpss

Description: A Cartesian product is included in the ordered pair universe. Exercise 3 of TakeutiZaring p. 25. (Contributed by NM, 2-Aug-1994)

	Ref	Expression
Assertion	xpss	$$⊢A\times B\subseteq \mathrm{V}\times \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		ssv	$$⊢A\subseteq \mathrm{V}$$
2		ssv	$$⊢B\subseteq \mathrm{V}$$
3		xpss12	$$⊢\left(A\subseteq \mathrm{V}\wedge B\subseteq \mathrm{V}\right)\to A\times B\subseteq \mathrm{V}\times \mathrm{V}$$
4	1 2 3	mp2an	$$⊢A\times B\subseteq \mathrm{V}\times \mathrm{V}$$