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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 ⟩ }