Theorem syl6ss 3011

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 3009	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:  difss2  3100  sstpr  3549  rintm  3765  eqbrrdva  4523  ssxpbm  4776  ssxp1  4777  ssxp2  4778  relfld  4866  funssxp  5080  dff2  5332  fliftf  5459  1stcof  5810  2ndcof  5811  tfrlemibfn  5965  sucinc2  6049  peano5nnnn  7058  peano5nni  8042  suprzclex  8445  ioodisj  9015  fzossnn0  9184  elfzom1elp1fzo  9211  frecuzrdgfn  9414  peano5set  10735