Theorem sstrd 2955

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

Hypotheses

Ref	Expression
sstrd.1	|- ( ph -> A C_ B )
sstrd.2	|- ( ph -> B C_ C )

Assertion

Ref	Expression
sstrd	|- ( ph -> A C_ C )

Proof of Theorem sstrd

Step	Hyp	Ref	Expression
1		sstrd.1	. 2 |- ( ph -> A C_ B )
2		sstrd.2	. 2 |- ( ph -> B C_ C )
3		sstr 2953	. 2 |- ( ( A C_ B /\ B C_ C ) -> A C_ C )
4	1, 2, 3	syl2anc 391	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by:  syl5ss  2956  syl6ss  2957  ssdif2d  3082  tfisi  4310  funss  4920  fssxp  5058  fvmptssdm  5255  suppssfv  5708  suppssov1  5709  tposss  5861  tfrlem1  5923  tfrlemibfn  5942  ecinxp  6181