Theorem sstri 3009

Description: Subclass transitivity inference. (Contributed by NM, 5-May-2000.)

Hypotheses

Ref	Expression
sstri.1	|- A C_ B
sstri.2	|- B C_ C

Assertion

Ref	Expression
sstri	|- A C_ C

Proof of Theorem sstri

Step	Hyp	Ref	Expression
1		sstri.1	. 2 |- A C_ B
2		sstri.2	. 2 |- B C_ C
3		sstr2 3007	. 2 |- ( A C_ B -> ( B C_ C -> A C_ C ) )
4	1, 2, 3	mp2 16	1 |- A C_ C

Syntax hints:   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:  difdif2ss  3228  difdifdirss  3334  snsstp1  3543  snsstp2  3544  nnregexmid  4368  dmexg  4624  rnexg  4625  ssrnres  4793  cossxp  4873  funinsn  4979  fabexg  5108  foimacnv  5175  ssimaex  5266  oprabss  5621  tposssxp  5898  dmaddpi  6577  dmmulpi  6578  ltrelxr  7240  nnsscn  8111  nn0sscn  8360  nn0ssq  8794  nnssq  8795  qsscn  8797  fzval2  9108  fzossnn  9275  fzo0ssnn0  9301  serige0  9570  expcl2lemap  9585  rpexpcl  9592  expge0  9609  expge1  9610  infssuzcldc  10491  isprm3  10644