Theorem lesid

Description: Surreal less-than or equal is reflexive. Theorem 0(iii) of Conway p. 16. (Contributed by Scott Fenton, 7-Aug-2024)

	Ref	Expression
Assertion	lesid	⊢ ( 𝐴 ∈ No → 𝐴 ≤s 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ltsirr	⊢ ( 𝐴 ∈ No → ¬ 𝐴 <s 𝐴 )
2		lenlts	⊢ ( ( 𝐴 ∈ No ∧ 𝐴 ∈ No ) → ( 𝐴 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐴 ) )
3	2	anidms	⊢ ( 𝐴 ∈ No → ( 𝐴 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐴 ) )
4	1 3	mpbird	⊢ ( 𝐴 ∈ No → 𝐴 ≤s 𝐴 )