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 × 𝐵 ) ⊆ ( V × V )

Proof

Step	Hyp	Ref	Expression
1		ssv	⊢ 𝐴 ⊆ V
2		ssv	⊢ 𝐵 ⊆ V
3		xpss12	⊢ ( ( 𝐴 ⊆ V ∧ 𝐵 ⊆ V ) → ( 𝐴 × 𝐵 ) ⊆ ( V × V ) )
4	1 2 3	mp2an	⊢ ( 𝐴 × 𝐵 ) ⊆ ( V × V )