Theorem slenegd

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

	Ref	Expression
Hypotheses	sltnegd.1	|- ( ph -> A e. No )
	sltnegd.2	|- ( ph -> B e. No )
Assertion	slenegd	|- ( ph -> ( A <_s B <-> ( -us ` B ) <_s ( -us ` A ) ) )

Proof

Step	Hyp	Ref	Expression
1		sltnegd.1	|- ( ph -> A e. No )
2		sltnegd.2	|- ( ph -> B e. No )
3		sleneg	|- ( ( A e. No /\ B e. No ) -> ( A <_s B <-> ( -us ` B ) <_s ( -us ` A ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( A <_s B <-> ( -us ` B ) <_s ( -us ` A ) ) )