Theorem sylan9ss 3993

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 3988	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)
4	1, 2, 3	syl2an 595	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-in 3954  df-ss 3964

This theorem is referenced by:  sylan9ssr  3994  psstr  4102  unss12  4182  ss2in  4237  ssdisj  4460  relrelss  6277  funssxp  6752  axdc3lem  10474  tskuni  10807  rtrclreclem4  15041  tsmsxp  24072  shslubi  31208  chlej12i  31298  insiga  33756  fnetr  35835  pcl0bN  39396  brtrclfv2  43157