Theorem subscld

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

	Ref	Expression
Hypotheses	subscld.1	$$⊢\phi \to A\in \mathrm{No}$$
	subscld.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	subscld	$$⊢\phi \to A{-}_{s}B\in \mathrm{No}$$

Proof

Step	Hyp	Ref	Expression
1		subscld.1	$$⊢\phi \to A\in \mathrm{No}$$
2		subscld.2	$$⊢\phi \to B\in \mathrm{No}$$
3		subscl	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to A{-}_{s}B\in \mathrm{No}$$
4	1 2 3	syl2anc	$$⊢\phi \to A{-}_{s}B\in \mathrm{No}$$