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Metamath Proof Explorer

Theorem onirri

Description: An ordinal number is not a member of itself. Theorem 7M(c) of Enderton p. 192. (Contributed by NM, 11-Jun-1994)

		Ref	Expression
	Hypothesis	on.1	$⊢ A \in On$
	Assertion	onirri	$⊢ \neg A \in A$

Step	Hyp	Ref	Expression
1		on.1	$⊢ A \in On$
2	1	onordi	$⊢ Ord A$
3		ordirr	$⊢ Ord A \to \neg A \in A$
4	2 3	ax-mp	$⊢ \neg A \in A$