Theorem epirron

Description: The strict order on the ordinals is irreflexive. Theorem 1.9(i) of Schloeder p. 1. (Contributed by RP, 15-Jan-2025)

	Ref	Expression
Assertion	epirron	$$⊢A\in \mathrm{On}\to \neg A\mathrm{E}A$$

Proof

Step	Hyp	Ref	Expression
1		epweon	$$⊢\mathrm{E}\mathrm{We}\mathrm{On}$$
2		weso	$$⊢\mathrm{E}\mathrm{We}\mathrm{On}\to \mathrm{E}\mathrm{Or}\mathrm{On}$$
3		sopo	$$⊢\mathrm{E}\mathrm{Or}\mathrm{On}\to \mathrm{E}\mathrm{Po}\mathrm{On}$$
4	1 2 3	mp2b	$$⊢\mathrm{E}\mathrm{Po}\mathrm{On}$$
5		poirr	$$⊢\left(\mathrm{E}\mathrm{Po}\mathrm{On}\wedge A\in \mathrm{On}\right)\to \neg A\mathrm{E}A$$
6	4 5	mpan	$$⊢A\in \mathrm{On}\to \neg A\mathrm{E}A$$