Theorem xpex

Description: The Cartesian product of two sets is a set. Proposition 6.2 of TakeutiZaring p. 23. (Contributed by NM, 14-Aug-1994)

	Ref	Expression
Hypotheses	xpex.1	⊢ 𝐴 ∈ V
	xpex.2	⊢ 𝐵 ∈ V
Assertion	xpex	⊢ ( 𝐴 × 𝐵 ) ∈ V

Proof

Step	Hyp	Ref	Expression
1		xpex.1	⊢ 𝐴 ∈ V
2		xpex.2	⊢ 𝐵 ∈ V
3		xpexg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 × 𝐵 ) ∈ V )
4	1 2 3	mp2an	⊢ ( 𝐴 × 𝐵 ) ∈ V