Theorem sletri3d

Description: Trichotomy law for surreal less-than or equal. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026)

	Ref	Expression
Hypotheses	sled.1	$$⊢\phi \to A\in \mathrm{No}$$
	sled.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	sletri3d	$$⊢\phi \to \left(A=B\leftrightarrow \left(A{\le }_{s}B\wedge B{\le }_{s}A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sled.1	$$⊢\phi \to A\in \mathrm{No}$$
2		sled.2	$$⊢\phi \to B\in \mathrm{No}$$
3		sletri3	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A=B\leftrightarrow \left(A{\le }_{s}B\wedge B{\le }_{s}A\right)\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(A=B\leftrightarrow \left(A{\le }_{s}B\wedge B{\le }_{s}A\right)\right)$$