Theorem subscld

Description: Closure law for surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	subscld.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	subscld.2	⊢ ( 𝜑 → 𝐵 ∈ No )
Assertion	subscld	⊢ ( 𝜑 → ( 𝐴 -s 𝐵 ) ∈ No )

Proof

Step	Hyp	Ref	Expression
1		subscld.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		subscld.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		subscl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 -s 𝐵 ) ∈ No )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 -s 𝐵 ) ∈ No )