Theorem gt0ne0sd

Description: A positive surreal is not equal to zero. (Contributed by Scott Fenton, 12-Mar-2025)

	Ref	Expression
Hypothesis	gt0ne0sd.1	$$⊢\phi \to 0_s{<}_{s}A$$
Assertion	gt0ne0sd	$$⊢\phi \to A\ne 0_s$$

Proof

Step	Hyp	Ref	Expression
1		gt0ne0sd.1	$$⊢\phi \to 0_s{<}_{s}A$$
2		gt0ne0s	$$⊢0_s{<}_{s}A\to A\ne 0_s$$
3	1 2	syl	$$⊢\phi \to A\ne 0_s$$