Theorem xpsnen2g

Description: A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014)

	Ref	Expression
Assertion	xpsnen2g	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left\{A\right\}\times B\right)\approx B$$

Proof

Step	Hyp	Ref	Expression
1		snex	$$⊢\left\{A\right\}\in \mathrm{V}$$
2		xpcomeng	$$⊢\left(\left\{A\right\}\in \mathrm{V}\wedge B\in W\right)\to \left(\left\{A\right\}\times B\right)\approx \left(B\times \left\{A\right\}\right)$$
3	1 2	mpan	$$⊢B\in W\to \left(\left\{A\right\}\times B\right)\approx \left(B\times \left\{A\right\}\right)$$
4		xpsneng	$$⊢\left(B\in W\wedge A\in V\right)\to \left(B\times \left\{A\right\}\right)\approx B$$
5	4	ancoms	$$⊢\left(A\in V\wedge B\in W\right)\to \left(B\times \left\{A\right\}\right)\approx B$$
6		entr	$$⊢\left(\left(\left\{A\right\}\times B\right)\approx \left(B\times \left\{A\right\}\right)\wedge \left(B\times \left\{A\right\}\right)\approx B\right)\to \left(\left\{A\right\}\times B\right)\approx B$$
7	3 5 6	syl2an2	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left\{A\right\}\times B\right)\approx B$$