Theorem ltlesd

Description: Surreal less-than implies less-than or equal. (Contributed by Scott Fenton, 16-Feb-2025)

	Ref	Expression
Hypotheses	ltlesd.1	$$⊢\phi \to A\in \mathrm{No}$$
	ltlesd.2	$$⊢\phi \to B\in \mathrm{No}$$
	ltlesd.3	$$⊢\phi \to A{<}_{s}B$$
Assertion	ltlesd	$$⊢\phi \to A{\le }_{s}B$$

Proof

Step	Hyp	Ref	Expression
1		ltlesd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		ltlesd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		ltlesd.3	$$⊢\phi \to A{<}_{s}B$$
4	1 2	jca	$$⊢\phi \to \left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)$$
5		ltsasym	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{<}_{s}B\to \neg B{<}_{s}A\right)$$
6	4 3 5	sylc	$$⊢\phi \to \neg B{<}_{s}A$$
7		lenlts	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{\le }_{s}B\leftrightarrow \neg B{<}_{s}A\right)$$
8	1 2 7	syl2anc	$$⊢\phi \to \left(A{\le }_{s}B\leftrightarrow \neg B{<}_{s}A\right)$$
9	6 8	mpbird	$$⊢\phi \to A{\le }_{s}B$$