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Theorem sstrd 3077
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
StepHypRef Expression
1 sstrd.1 . 2  |-  ( ph  ->  A  C_  B )
2 sstrd.2 . 2  |-  ( ph  ->  B  C_  C )
3 sstr 3075 . 2  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
41, 2, 3syl2anc 408 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-in 3047  df-ss 3054
This theorem is referenced by:  sstrid  3078  sstrdi  3079  ssdif2d  3185  tfisi  4471  funss  5112  fssxp  5260  fvmptssdm  5473  suppssfv  5946  suppssov1  5947  tposss  6111  tfrlem1  6173  tfrlemibfn  6193  tfr1onlembfn  6209  tfr1onlemubacc  6211  tfr1onlemres  6214  tfrcllembfn  6222  tfrcllemubacc  6224  tfrcllemres  6227  ecinxp  6472  undifdc  6780  sbthlem1  6813  iseqf1olemnab  10216  isumss  11115  ennnfoneleminc  11835  strsetsid  11903  strleund  11958  ntrss  12199  neiint  12225  neiss  12230  restopnb  12261  iscnp4  12298  blssps  12507  blss  12508  xmettx  12590  tgqioo  12627  rescncf  12648  suplociccreex  12682  suplociccex  12683  dvbss  12734  dvbsssg  12735  dvfgg  12737  dvcnp2cntop  12743  dvcn  12744  dvaddxxbr  12745  dvmulxxbr  12746  dvcoapbr  12751
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