Theorem maxs1

Description: A surreal is less than or equal to the maximum of it and another. (Contributed by Scott Fenton, 14-Feb-2025)

	Ref	Expression
Assertion	maxs1	$$⊢A\in \mathrm{No}\to A{\le }_{s}if\left(A{\le }_{s}B,B,A\right)$$

Proof

Step	Hyp	Ref	Expression
1		lesid	$$⊢A\in \mathrm{No}\to A{\le }_{s}A$$
2		iffalse	$$⊢\neg A{\le }_{s}B\to if\left(A{\le }_{s}B,B,A\right)=A$$
3	2	breq2d	$$⊢\neg A{\le }_{s}B\to \left(A{\le }_{s}if\left(A{\le }_{s}B,B,A\right)\leftrightarrow A{\le }_{s}A\right)$$
4	1 3	syl5ibrcom	$$⊢A\in \mathrm{No}\to \left(\neg A{\le }_{s}B\to A{\le }_{s}if\left(A{\le }_{s}B,B,A\right)\right)$$
5		id	$$⊢A{\le }_{s}B\to A{\le }_{s}B$$
6		iftrue	$$⊢A{\le }_{s}B\to if\left(A{\le }_{s}B,B,A\right)=B$$
7	5 6	breqtrrd	$$⊢A{\le }_{s}B\to A{\le }_{s}if\left(A{\le }_{s}B,B,A\right)$$
8	4 7	pm2.61d2	$$⊢A\in \mathrm{No}\to A{\le }_{s}if\left(A{\le }_{s}B,B,A\right)$$