Theorem sletrd

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

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

Proof

Step	Hyp	Ref	Expression
1		slttrd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		slttrd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		slttrd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		sletrd.4	⊢ ( 𝜑 → 𝐴 ≤s 𝐵 )
5		sletrd.5	⊢ ( 𝜑 → 𝐵 ≤s 𝐶 )
6		sletr	⊢ ( ( 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 )