Theorem sylan9ss 3944

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 3939	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)
4	1, 2, 3	syl2an 608	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ⊆ wss 3899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ss 3916

This theorem is used by:  sylan9ssr  3945  psstr  4056  unss12  4134  ss2in  4190  ssdisj  4413  relrelss  6270  funssxp  6731  axdc3lem  10452  tskuni  10792  rtrclreclem4  15134  tsmsxp  24381  shslubi  31866  chlej12i  31956  insiga  34648  fnetr  36970  pcl0bN  40796  brtrclfv2  44567