Theorem scutcl

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

	Ref	Expression
Assertion	scutcl	⊢ ( 𝐴 <<s 𝐵 → ( 𝐴 |s 𝐵 ) ∈ No )

Proof

Step	Hyp	Ref	Expression
1		scutcut	⊢ ( 𝐴 <<s 𝐵 → ( ( 𝐴 |s 𝐵 ) ∈ No ∧ 𝐴 <<s { ( 𝐴 |s 𝐵 ) } ∧ { ( 𝐴 |s 𝐵 ) } <<s 𝐵 ) )
2	1	simp1d	⊢ ( 𝐴 <<s 𝐵 → ( 𝐴 |s 𝐵 ) ∈ No )