Theorem xpexd

Description: The Cartesian product of two sets is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	xpexd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	xpexd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
Assertion	xpexd	⊢ ( 𝜑 → ( 𝐴 × 𝐵 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		xpexd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		xpexd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3		xpexg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 × 𝐵 ) ∈ V )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 × 𝐵 ) ∈ V )