Theorem xpcomen

Description: Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of Mendelson p. 254. (Contributed by NM, 5-Jan-2004) (Revised by Mario Carneiro, 15-Nov-2014)

	Ref	Expression
Hypotheses	xpcomen.1	$$⊢A\in \mathrm{V}$$
	xpcomen.2	$$⊢B\in \mathrm{V}$$
Assertion	xpcomen	$$⊢\left(A\times B\right)\approx \left(B\times A\right)$$

Proof

Step	Hyp	Ref	Expression
1		xpcomen.1	$$⊢A\in \mathrm{V}$$
2		xpcomen.2	$$⊢B\in \mathrm{V}$$
3	1 2	xpex	$$⊢A\times B\in \mathrm{V}$$
4	2 1	xpex	$$⊢B\times A\in \mathrm{V}$$
5		eqid	$$⊢\left(x\in \left(A\times B\right)⟼\bigcup {\left\{x\right\}}^{-1}\right)=\left(x\in \left(A\times B\right)⟼\bigcup {\left\{x\right\}}^{-1}\right)$$
6	5	xpcomf1o	$$⊢\left(x\in \left(A\times B\right)⟼\bigcup {\left\{x\right\}}^{-1}\right):A\times B\underset{1-1\; onto}{⟶}B\times A$$
7		f1oen2g	$$⊢\left(A\times B\in \mathrm{V}\wedge B\times A\in \mathrm{V}\wedge \left(x\in \left(A\times B\right)⟼\bigcup {\left\{x\right\}}^{-1}\right):A\times B\underset{1-1\; onto}{⟶}B\times A\right)\to \left(A\times B\right)\approx \left(B\times A\right)$$
8	3 4 6 7	mp3an	$$⊢\left(A\times B\right)\approx \left(B\times A\right)$$