Theorem lestrd

Description: Surreal less-than or equal is transitive. (Contributed by Scott Fenton, 8-Dec-2021)

	Ref	Expression
Hypotheses	ltstrd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	ltstrd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	ltstrd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
	lestrd.4	⊢ ( 𝜑 → 𝐴 ≤s 𝐵 )
	lestrd.5	⊢ ( 𝜑 → 𝐵 ≤s 𝐶 )
Assertion	lestrd	⊢ ( 𝜑 → 𝐴 ≤s 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ltstrd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		ltstrd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		ltstrd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		lestrd.4	⊢ ( 𝜑 → 𝐴 ≤s 𝐵 )
5		lestrd.5	⊢ ( 𝜑 → 𝐵 ≤s 𝐶 )
6		lestr	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶 ) → 𝐴 ≤s 𝐶 ) )
7	1 2 3 6	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶 ) → 𝐴 ≤s 𝐶 ) )
8	4 5 7	mp2and	⊢ ( 𝜑 → 𝐴 ≤s 𝐶 )