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Metamath Proof Explorer

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 )