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 ⟩ }