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  6274  funssxp  6736  axdc3lem  10521  tskuni  10861  rtrclreclem4  15207  tsmsxp  24467  shslubi  31980  chlej12i  32070  insiga  34763  fnetr  37119  pcl0bN  40960  brtrclfv2  44712