Theorem sylan9ss

Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004) (Proof shortened by Andrew Salmon, 14-Jun-2011)

	Ref	Expression
Hypotheses	sylan9ss.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	sylan9ss.2	⊢ ( 𝜓 → 𝐵 ⊆ 𝐶 )
Assertion	sylan9ss	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sylan9ss.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		sylan9ss.2	⊢ ( 𝜓 → 𝐵 ⊆ 𝐶 )
3		sstr	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶 ) → 𝐴 ⊆ 𝐶 )
4	1 2 3	syl2an	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 ⊆ 𝐶 )