Theorem iprc

Description: The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set, as in idcn . (Contributed by NM, 1-Jan-2007)

	Ref	Expression
Assertion	iprc	$$⊢\neg \mathrm{I}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		dmi	$$⊢\mathrm{dom}\mathrm{I}=\mathrm{V}$$
2		vprc	$$⊢\neg \mathrm{V}\in \mathrm{V}$$
3	1 2	eqneltri	$$⊢\neg \mathrm{dom}\mathrm{I}\in \mathrm{V}$$
4		dmexg	$$⊢\mathrm{I}\in \mathrm{V}\to \mathrm{dom}\mathrm{I}\in \mathrm{V}$$
5	3 4	mto	$$⊢\neg \mathrm{I}\in \mathrm{V}$$