Theorem syl6ss 3012

Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
syl6ss.1	|- ( ph -> A C_ B )
syl6ss.2	|- B C_ C

Assertion

Ref	Expression
syl6ss	|- ( ph -> A C_ C )

Proof of Theorem syl6ss

Step	Hyp	Ref	Expression
1		syl6ss.1	. 2 |- ( ph -> A C_ B )
2		syl6ss.2	. . 3 |- B C_ C
3	2	a1i 9	. 2 |- ( ph -> B C_ C )
4	1, 3	sstrd 3010	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   C_ wss 2974

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987

This theorem is referenced by:  difss2  3101  sstpr  3557  rintm  3773  eqbrrdva  4533  ssxpbm  4786  ssxp1  4787  ssxp2  4788  relfld  4876  funssxp  5091  dff2  5343  fliftf  5470  1stcof  5821  2ndcof  5822  tfrlemibfn  5977  tfr1onlembfn  5993  tfrcllemssrecs  6001  tfrcllembfn  6006  sucinc2  6090  peano5nnnn  7120  peano5nni  8109  suprzclex  8526  ioodisj  9091  fzossnn0  9261  elfzom1elp1fzo  9288  frecuzrdgtcl  9494  frecuzrdgdomlem  9499  frecuzrdgfunlem  9501  peano5set  10893