Theorem sstrd 3036

Description: Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)

Hypotheses

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

Assertion

Ref	Expression
sstrd	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem sstrd

Step	Hyp	Ref	Expression
1		sstrd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sstrd.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
3		sstr 3034	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)
4	1, 2, 3	syl2anc 404	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 3000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-in 3006  df-ss 3013

This theorem is referenced by:  syl5ss  3037  syl6ss  3038  ssdif2d  3140  tfisi  4415  funss  5047  fssxp  5191  fvmptssdm  5400  suppssfv  5866  suppssov1  5867  tposss  6025  tfrlem1  6087  tfrlemibfn  6107  tfr1onlembfn  6123  tfr1onlemubacc  6125  tfr1onlemres  6128  tfrcllembfn  6136  tfrcllemubacc  6138  tfrcllemres  6141  ecinxp  6381  undifdc  6688  sbthlem1  6720  iseqf1olemnab  9978  isumss  10844  strsetsid  11588  strleund  11643  ntrss  11880  rescncf  11910