Theorem negsex

Description: Every surreal has a negative. Note that this theorem, addscl , addscom , addsass , addsrid , and sltadd1im are the ordered Abelian group axioms. However, the surreals cannot be said to be an ordered Abelian group because No is a proper class. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	negsex	|- ( A e. No -> E. x e. No ( A +s x ) = 0s )

Proof

Step	Hyp	Ref	Expression
1		negscl	|- ( A e. No -> ( -us ` A ) e. No )
2		negsid	|- ( A e. No -> ( A +s ( -us ` A ) ) = 0s )
3		oveq2	|- ( x = ( -us ` A ) -> ( A +s x ) = ( A +s ( -us ` A ) ) )
4	3	eqeq1d	|- ( x = ( -us ` A ) -> ( ( A +s x ) = 0s <-> ( A +s ( -us ` A ) ) = 0s ) )
5	4	rspcev	|- ( ( ( -us ` A ) e. No /\ ( A +s ( -us ` A ) ) = 0s ) -> E. x e. No ( A +s x ) = 0s )
6	1 2 5	syl2anc	|- ( A e. No -> E. x e. No ( A +s x ) = 0s )