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 ∈ No → 𝐴 ≤s if ( 𝐴 ≤s 𝐵 , 𝐵 , 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		slerflex	⊢ ( 𝐴 ∈ No → 𝐴 ≤s 𝐴 )
2		iffalse	⊢ ( ¬ 𝐴 ≤s 𝐵 → if ( 𝐴 ≤s 𝐵 , 𝐵 , 𝐴 ) = 𝐴 )
3	2	breq2d	⊢ ( ¬ 𝐴 ≤s 𝐵 → ( 𝐴 ≤s if ( 𝐴 ≤s 𝐵 , 𝐵 , 𝐴 ) ↔ 𝐴 ≤s 𝐴 ) )
4	1 3	syl5ibrcom	⊢ ( 𝐴 ∈ No → ( ¬ 𝐴 ≤s 𝐵 → 𝐴 ≤s if ( 𝐴 ≤s 𝐵 , 𝐵 , 𝐴 ) ) )
5		id	⊢ ( 𝐴 ≤s 𝐵 → 𝐴 ≤s 𝐵 )
6		iftrue	⊢ ( 𝐴 ≤s 𝐵 → if ( 𝐴 ≤s 𝐵 , 𝐵 , 𝐴 ) = 𝐵 )
7	5 6	breqtrrd	⊢ ( 𝐴 ≤s 𝐵 → 𝐴 ≤s if ( 𝐴 ≤s 𝐵 , 𝐵 , 𝐴 ) )
8	4 7	pm2.61d2	⊢ ( 𝐴 ∈ No → 𝐴 ≤s if ( 𝐴 ≤s 𝐵 , 𝐵 , 𝐴 ) )