Theorem sstrd 3009

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 3007	. 2 |- ( ( A C_ B /\ B C_ C ) -> A C_ C )
4	1, 2, 3	syl2anc 403	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  syl5ss  3010  syl6ss  3011  ssdif2d  3111  tfisi  4328  funss  4940  fssxp  5078  fvmptssdm  5276  suppssfv  5728  suppssov1  5729  tposss  5884  tfrlem1  5946  tfrlemibfn  5965  ecinxp  6204