Theorem ltnegsd

Description: Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 14-Mar-2025)

	Ref	Expression
Hypotheses	ltnegsd.1	$$⊢\phi \to A\in \mathrm{No}$$
	ltnegsd.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	ltnegsd	$$⊢\phi \to \left(A{<}_{s}B\leftrightarrow {+}_s\left(B\right){<}_s{+}_s\left(A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltnegsd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		ltnegsd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		ltnegs	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{<}_{s}B\leftrightarrow {+}_s\left(B\right){<}_s{+}_s\left(A\right)\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(A{<}_{s}B\leftrightarrow {+}_s\left(B\right){<}_s{+}_s\left(A\right)\right)$$