Theorem sltasym

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

	Ref	Expression
Assertion	sltasym	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{<}_{s}B\to \neg B{<}_{s}A\right)$$

Proof

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