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Metamath Proof Explorer

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	⊢ ¬ I ∈ V

Proof

Step	Hyp	Ref	Expression
1		dmi	⊢ dom I = V
2		vprc	⊢ ¬ V ∈ V
3	1 2	eqneltri	⊢ ¬ dom I ∈ V
4		dmexg	⊢ ( I ∈ V → dom I ∈ V )
5	3 4	mto	⊢ ¬ I ∈ V