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	$$⊢A\in \mathrm{On}$$
Assertion	onirri	$$⊢\neg A\in A$$

Proof

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