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	Could not format assertion : No typesetting found for |- ( A e. No -> E. x e. No ( A +s x ) = 0s ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		negscl	Could not format ( A e. No -> ( -us ` A ) e. No ) : No typesetting found for |- ( A e. No -> ( -us ` A ) e. No ) with typecode |-
2		negsid	Could not format ( A e. No -> ( A +s ( -us ` A ) ) = 0s ) : No typesetting found for |- ( A e. No -> ( A +s ( -us ` A ) ) = 0s ) with typecode |-
3		oveq2	Could not format ( x = ( -us ` A ) -> ( A +s x ) = ( A +s ( -us ` A ) ) ) : No typesetting found for |- ( x = ( -us ` A ) -> ( A +s x ) = ( A +s ( -us ` A ) ) ) with typecode |-
4	3	eqeq1d	Could not format ( x = ( -us ` A ) -> ( ( A +s x ) = 0s <-> ( A +s ( -us ` A ) ) = 0s ) ) : No typesetting found for |- ( x = ( -us ` A ) -> ( ( A +s x ) = 0s <-> ( A +s ( -us ` A ) ) = 0s ) ) with typecode |-
5	4	rspcev	Could not format ( ( ( -us ` A ) e. No /\ ( A +s ( -us ` A ) ) = 0s ) -> E. x e. No ( A +s x ) = 0s ) : No typesetting found for |- ( ( ( -us ` A ) e. No /\ ( A +s ( -us ` A ) ) = 0s ) -> E. x e. No ( A +s x ) = 0s ) with typecode |-
6	1 2 5	syl2anc	Could not format ( A e. No -> E. x e. No ( A +s x ) = 0s ) : No typesetting found for |- ( A e. No -> E. x e. No ( A +s x ) = 0s ) with typecode |-