Theorem sylan9ss 3951

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

Hypotheses

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

Assertion

Ref	Expression
sylan9ss	⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶)

Proof of Theorem sylan9ss

Step	Hyp	Ref	Expression
1		sylan9ss.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sylan9ss.2	. 2 ⊢ (𝜓 → 𝐵 ⊆ 𝐶)
3		sstr 3946	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)
4	1, 2, 3	syl2an 596	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ss 3922

This theorem is referenced by:  sylan9ssr  3952  psstr  4060  unss12  4141  ss2in  4198  ssdisj  4413  relrelss  6225  funssxp  6684  axdc3lem  10363  tskuni  10696  rtrclreclem4  14987  tsmsxp  24059  shslubi  31348  chlej12i  31438  insiga  34123  fnetr  36344  pcl0bN  39922  brtrclfv2  43720