Theorem sleneg

Description: Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	sleneg	Could not format assertion : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( A <_s B <-> ( -us ` B ) <_s ( -us ` A ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		sltneg	Could not format ( ( B e. No /\ A e. No ) -> ( B ( -us ` A ) ( B ( -us ` A )
2	1	ancoms	Could not format ( ( A e. No /\ B e. No ) -> ( B ( -us ` A ) ( B ( -us ` A )
3	2	notbid	Could not format ( ( A e. No /\ B e. No ) -> ( -. B -. ( -us ` A ) ( -. B -. ( -us ` A )
4		slenlt	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{\le }_{s}B\leftrightarrow \neg B{<}_{s}A\right)$$
5		negscl	Could not format ( B e. No -> ( -us ` B ) e. No ) : No typesetting found for |- ( B e. No -> ( -us ` B ) e. No ) with typecode |-
6		negscl	Could not format ( A e. No -> ( -us ` A ) e. No ) : No typesetting found for |- ( A e. No -> ( -us ` A ) e. No ) with typecode |-
7		slenlt	Could not format ( ( ( -us ` B ) e. No /\ ( -us ` A ) e. No ) -> ( ( -us ` B ) <_s ( -us ` A ) <-> -. ( -us ` A ) ( ( -us ` B ) <_s ( -us ` A ) <-> -. ( -us ` A )
8	5 6 7	syl2anr	Could not format ( ( A e. No /\ B e. No ) -> ( ( -us ` B ) <_s ( -us ` A ) <-> -. ( -us ` A ) ( ( -us ` B ) <_s ( -us ` A ) <-> -. ( -us ` A )
9	3 4 8	3bitr4d	Could not format ( ( A e. No /\ B e. No ) -> ( A <_s B <-> ( -us ` B ) <_s ( -us ` A ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( A <_s B <-> ( -us ` B ) <_s ( -us ` A ) ) ) with typecode |-