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Theorem sstri 3034
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
StepHypRef Expression
1 sstri.1 . 2  |-  A  C_  B
2 sstri.2 . 2  |-  B  C_  C
3 sstr2 3032 . 2  |-  ( A 
C_  B  ->  ( B  C_  C  ->  A  C_  C ) )
41, 2, 3mp2 16 1  |-  A  C_  C
Colors of variables: wff set class
Syntax hints:    C_ wss 2999
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012
This theorem is referenced by:  difdif2ss  3256  difdifdirss  3367  snsstp1  3587  snsstp2  3588  nnregexmid  4434  dmexg  4697  rnexg  4698  ssrnres  4873  cossxp  4953  cocnvss  4956  funinsn  5063  fabexg  5198  foimacnv  5271  ssimaex  5365  oprabss  5734  tposssxp  6014  mapsspw  6439  sbthlemi5  6668  sbthlem7  6670  djuin  6754  caserel  6776  dmaddpi  6882  dmmulpi  6883  ltrelxr  7545  nnsscn  8425  nn0sscn  8676  nn0ssq  9111  nnssq  9112  qsscn  9114  fzval2  9425  fzossnn  9596  fzo0ssnn0  9622  expcl2lemap  9963  rpexpcl  9970  expge0  9987  expge1  9988  iseqcoll  10243  isummolem2a  10767  fsum3cvg3  10785  fsumrpcl  10794  fsumge0  10849  infssuzcldc  11221  isprm3  11374  structfn  11509  toponsspwpwg  11573  dmtopon  11574
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