Theorem lt0negs2d

Description: Comparison of a surreal and its negative to zero. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypothesis	lt0negs2d.1	|- ( ph -> A e. No )
Assertion	lt0negs2d	|- ( ph -> ( 0s ( -us ` A )

Proof

Step	Hyp	Ref	Expression
1		lt0negs2d.1	|- ( ph -> A e. No )
2		0no	|- 0s e. No
3		ltnegs	|- ( ( 0s e. No /\ A e. No ) -> ( 0s ( -us ` A )
4	2 1 3	sylancr	|- ( ph -> ( 0s ( -us ` A )
5		neg0s	|- ( -us ` 0s ) = 0s
6	5	breq2i	|- ( ( -us ` A ) ( -us ` A )
7	4 6	bitrdi	|- ( ph -> ( 0s ( -us ` A )