Theorem nordeq

Description: A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998)

	Ref	Expression
Assertion	nordeq	$$⊢\left(\mathrm{Ord}A\wedge B\in A\right)\to A\ne B$$

Proof

Step	Hyp	Ref	Expression
1		ordirr	$$⊢\mathrm{Ord}A\to \neg A\in A$$
2		eleq1	$$⊢A=B\to \left(A\in A\leftrightarrow B\in A\right)$$
3	2	notbid	$$⊢A=B\to \left(\neg A\in A\leftrightarrow \neg B\in A\right)$$
4	1 3	syl5ibcom	$$⊢\mathrm{Ord}A\to \left(A=B\to \neg B\in A\right)$$
5	4	necon2ad	$$⊢\mathrm{Ord}A\to \left(B\in A\to A\ne B\right)$$
6	5	imp	$$⊢\left(\mathrm{Ord}A\wedge B\in A\right)\to A\ne B$$