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	$$⊢A\in \mathrm{V}$$
	xpsn.2	$$⊢B\in \mathrm{V}$$
Assertion	xpsn	$$⊢\left\{A\right\}\times \left\{B\right\}=\left\{⟨A,B⟩\right\}$$

Proof

Step	Hyp	Ref	Expression
1		xpsn.1	$$⊢A\in \mathrm{V}$$
2		xpsn.2	$$⊢B\in \mathrm{V}$$
3		xpsng	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left\{A\right\}\times \left\{B\right\}=\left\{⟨A,B⟩\right\}$$
4	1 2 3	mp2an	$$⊢\left\{A\right\}\times \left\{B\right\}=\left\{⟨A,B⟩\right\}$$