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Metamath Proof Explorer

Theorem xpsn

Description: The Cartesian product of two singletons is the singleton consisting in the associated ordered pair. (Contributed by NM, 4-Nov-2006)

Ref Expression
Hypotheses xpsn.1 ⊢ A ∈ V
xpsn.2 ⊢ B ∈ V
Assertion xpsn ⊢ A × B = A B

Proof

Step Hyp Ref Expression
1 xpsn.1 ⊢ A ∈ V
2 xpsn.2 ⊢ B ∈ V
3 xpsng ⊢ A ∈ V ∧ B ∈ V → A × B = A B
4 1 2 3 mp2an ⊢ A × B = A B