Theorem predfrirr

Description: Given a well-founded relation, a class is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011)

	Ref	Expression
Assertion	predfrirr	$$⊢R\mathrm{Fr}A\to \neg X\in Pred\left(R,A,X\right)$$

Proof

Step	Hyp	Ref	Expression
1		frirr	$$⊢\left(R\mathrm{Fr}A\wedge X\in A\right)\to \neg XRX$$
2		elpredg	$$⊢\left(X\in A\wedge X\in A\right)\to \left(X\in Pred\left(R,A,X\right)\leftrightarrow XRX\right)$$
3	2	anidms	$$⊢X\in A\to \left(X\in Pred\left(R,A,X\right)\leftrightarrow XRX\right)$$
4	3	notbid	$$⊢X\in A\to \left(\neg X\in Pred\left(R,A,X\right)\leftrightarrow \neg XRX\right)$$
5	1 4	syl5ibr	$$⊢X\in A\to \left(\left(R\mathrm{Fr}A\wedge X\in A\right)\to \neg X\in Pred\left(R,A,X\right)\right)$$
6	5	expd	$$⊢X\in A\to \left(R\mathrm{Fr}A\to \left(X\in A\to \neg X\in Pred\left(R,A,X\right)\right)\right)$$
7	6	pm2.43b	$$⊢R\mathrm{Fr}A\to \left(X\in A\to \neg X\in Pred\left(R,A,X\right)\right)$$
8		predel	$$⊢X\in Pred\left(R,A,X\right)\to X\in A$$
9	8	con3i	$$⊢\neg X\in A\to \neg X\in Pred\left(R,A,X\right)$$
10	7 9	pm2.61d1	$$⊢R\mathrm{Fr}A\to \neg X\in Pred\left(R,A,X\right)$$