Theorem mins2

Description: The minimum of two surreals is less than or equal to the second. (Contributed by Scott Fenton, 14-Feb-2025)

	Ref	Expression
Assertion	mins2	⊢ ( 𝐵 ∈ No → if ( 𝐴 ≤s 𝐵 , 𝐴 , 𝐵 ) ≤s 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		lesid	⊢ ( 𝐵 ∈ No → 𝐵 ≤s 𝐵 )
2		iffalse	⊢ ( ¬ 𝐴 ≤s 𝐵 → if ( 𝐴 ≤s 𝐵 , 𝐴 , 𝐵 ) = 𝐵 )
3	2	breq1d	⊢ ( ¬ 𝐴 ≤s 𝐵 → ( if ( 𝐴 ≤s 𝐵 , 𝐴 , 𝐵 ) ≤s 𝐵 ↔ 𝐵 ≤s 𝐵 ) )
4	1 3	syl5ibrcom	⊢ ( 𝐵 ∈ No → ( ¬ 𝐴 ≤s 𝐵 → if ( 𝐴 ≤s 𝐵 , 𝐴 , 𝐵 ) ≤s 𝐵 ) )
5		iftrue	⊢ ( 𝐴 ≤s 𝐵 → if ( 𝐴 ≤s 𝐵 , 𝐴 , 𝐵 ) = 𝐴 )
6		id	⊢ ( 𝐴 ≤s 𝐵 → 𝐴 ≤s 𝐵 )
7	5 6	eqbrtrd	⊢ ( 𝐴 ≤s 𝐵 → if ( 𝐴 ≤s 𝐵 , 𝐴 , 𝐵 ) ≤s 𝐵 )
8	4 7	pm2.61d2	⊢ ( 𝐵 ∈ No → if ( 𝐴 ≤s 𝐵 , 𝐴 , 𝐵 ) ≤s 𝐵 )