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Metamath Proof Explorer

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