Theorem negsex

Description: Every surreal has a negative. Note that this theorem, addscl , addscom , addsass , addsrid , and ltadds1im 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 ∈ No → ∃ 𝑥 ∈ No ( 𝐴 +s 𝑥 ) = 0s )

Proof

Step	Hyp	Ref	Expression
1		negscl	⊢ ( 𝐴 ∈ No → ( -us ‘ 𝐴 ) ∈ No )
2		negsid	⊢ ( 𝐴 ∈ 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 ) ∈ No ∧ ( 𝐴 +s ( -us ‘ 𝐴 ) ) = 0s ) → ∃ 𝑥 ∈ No ( 𝐴 +s 𝑥 ) = 0s )
6	1 2 5	syl2anc	⊢ ( 𝐴 ∈ No → ∃ 𝑥 ∈ No ( 𝐴 +s 𝑥 ) = 0s )