Theorem subseq0d

Description: The difference between two surreals is zero iff they are equal. (Contributed by Scott Fenton, 7-Nov-2025)

	Ref	Expression
Hypotheses	subseq0d.1	$$⊢\phi \to A\in \mathrm{No}$$
	subseq0d.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	subseq0d	$$⊢\phi \to \left(A{-}_{s}B=0_s\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		subseq0d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		subseq0d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		subsid	$$⊢B\in \mathrm{No}\to B{-}_{s}B=0_s$$
4	2 3	syl	$$⊢\phi \to B{-}_{s}B=0_s$$
5	4	eqeq2d	$$⊢\phi \to \left(A{-}_{s}B=B{-}_{s}B\leftrightarrow A{-}_{s}B=0_s\right)$$
6	1 2 2	subscan2d	$$⊢\phi \to \left(A{-}_{s}B=B{-}_{s}B\leftrightarrow A=B\right)$$
7	5 6	bitr3d	$$⊢\phi \to \left(A{-}_{s}B=0_s\leftrightarrow A=B\right)$$