Theorem sltnle

Description: Surreal less-than in terms of less-than or equal. (Contributed by Scott Fenton, 8-Dec-2021)

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

Proof

Step	Hyp	Ref	Expression
1		slenlt	$$⊢\left(B\in \mathrm{No}\wedge A\in \mathrm{No}\right)\to \left(B{\le }_{s}A\leftrightarrow \neg A{<}_{s}B\right)$$
2	1	ancoms	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(B{\le }_{s}A\leftrightarrow \neg A{<}_{s}B\right)$$
3	2	con2bid	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{<}_{s}B\leftrightarrow \neg B{\le }_{s}A\right)$$