Theorem sylan9ss 3963

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 3958	. 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 3917

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 3934

This theorem is referenced by:  sylan9ssr  3964  psstr  4073  unss12  4154  ss2in  4211  ssdisj  4426  relrelss  6249  funssxp  6719  axdc3lem  10410  tskuni  10743  rtrclreclem4  15034  tsmsxp  24049  shslubi  31321  chlej12i  31411  insiga  34134  fnetr  36346  pcl0bN  39924  brtrclfv2  43723