Theorem xp0

Description: The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of Monk1 p. 37. (Contributed by NM, 12-Apr-2004)

	Ref	Expression
Assertion	xp0	$$⊢A\times \varnothing =\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		0xp	$$⊢\varnothing \times A=\varnothing$$
2	1	cnveqi	$$⊢{\left(\varnothing \times A\right)}^{-1}={\varnothing }^{-1}$$
3		cnvxp	$$⊢{\left(\varnothing \times A\right)}^{-1}=A\times \varnothing$$
4		cnv0	$$⊢{\varnothing }^{-1}=\varnothing$$
5	2 3 4	3eqtr3i	$$⊢A\times \varnothing =\varnothing$$