This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )