Theorem negscld

Description: The surreals are closed under negation. Theorem 6(ii) of Conway p. 18. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Hypothesis	negscld.1	⊢ ( 𝜑 → 𝐴 ∈ No )
Assertion	negscld	⊢ ( 𝜑 → ( -us ‘ 𝐴 ) ∈ No )

Proof

Step	Hyp	Ref	Expression
1		negscld.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		negscl	⊢ ( 𝐴 ∈ No → ( -us ‘ 𝐴 ) ∈ No )
3	1 2	syl	⊢ ( 𝜑 → ( -us ‘ 𝐴 ) ∈ No )