Theorem negscl

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

	Ref	Expression
Assertion	negscl	⊢ ( 𝐴 ∈ No → ( -us ‘ 𝐴 ) ∈ No )

Proof

Step	Hyp	Ref	Expression
1		0sno	⊢ 0s ∈ No
2		negsprop	⊢ ( ( 𝐴 ∈ No ∧ 0s ∈ No ) → ( ( -us ‘ 𝐴 ) ∈ No ∧ ( 𝐴 <s 0s → ( -us ‘ 0s ) <s ( -us ‘ 𝐴 ) ) ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ No → ( ( -us ‘ 𝐴 ) ∈ No ∧ ( 𝐴 <s 0s → ( -us ‘ 0s ) <s ( -us ‘ 𝐴 ) ) ) )
4	3	simpld	⊢ ( 𝐴 ∈ No → ( -us ‘ 𝐴 ) ∈ No )