Theorem cutscld

Description: Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024)

	Ref	Expression
Hypothesis	cutscld.1	⊢ ( 𝜑 → 𝐴 <<s 𝐵 )
Assertion	cutscld	⊢ ( 𝜑 → ( 𝐴 |s 𝐵 ) ∈ No )

Proof

Step	Hyp	Ref	Expression
1		cutscld.1	⊢ ( 𝜑 → 𝐴 <<s 𝐵 )
2		cutscl	⊢ ( 𝐴 <<s 𝐵 → ( 𝐴 |s 𝐵 ) ∈ No )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐴 |s 𝐵 ) ∈ No )