Theorem ltsirr

Description: Surreal less-than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011)

	Ref	Expression
Assertion	ltsirr	$$⊢A\in \mathrm{No}\to \neg A{<}_{s}A$$

Proof

Step	Hyp	Ref	Expression
1		ltsso	$$⊢{<}_s\mathrm{Or}\mathrm{No}$$
2		sonr	$$⊢\left({<}_s\mathrm{Or}\mathrm{No}\wedge A\in \mathrm{No}\right)\to \neg A{<}_{s}A$$
3	1 2	mpan	$$⊢A\in \mathrm{No}\to \neg A{<}_{s}A$$