Theorem sltne

Description: Surreal less-than implies not equal. (Contributed by Scott Fenton, 12-Mar-2025)

	Ref	Expression
Assertion	sltne	$$⊢\left(A\in \mathrm{No}\wedge A{<}_{s}B\right)\to B\ne A$$

Proof

Step	Hyp	Ref	Expression
1		sltirr	$$⊢A\in \mathrm{No}\to \neg A{<}_{s}A$$
2		breq2	$$⊢B=A\to \left(A{<}_{s}B\leftrightarrow A{<}_{s}A\right)$$
3	2	notbid	$$⊢B=A\to \left(\neg A{<}_{s}B\leftrightarrow \neg A{<}_{s}A\right)$$
4	1 3	syl5ibrcom	$$⊢A\in \mathrm{No}\to \left(B=A\to \neg A{<}_{s}B\right)$$
5	4	necon2ad	$$⊢A\in \mathrm{No}\to \left(A{<}_{s}B\to B\ne A\right)$$
6	5	imp	$$⊢\left(A\in \mathrm{No}\wedge A{<}_{s}B\right)\to B\ne A$$