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Theorem ordtr1 4206
Description: Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
Assertion
Ref Expression
ordtr1  |-  ( Ord 
C  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )

Proof of Theorem ordtr1
StepHypRef Expression
1 ordtr 4196 . 2  |-  ( Ord 
C  ->  Tr  C
)
2 trel 3935 . 2  |-  ( Tr  C  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )
31, 2syl 14 1  |-  ( Ord 
C  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   Tr wtr 3928   Ord word 4180
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  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-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-uni 3649  df-tr 3929  df-iord 4184
This theorem is referenced by:  ontr1  4207  ordwe  4381  dfsmo2  6034  smores2  6041  smoel  6047  tfr1onlemsucaccv  6088  tfr1onlembxssdm  6090  tfr1onlembfn  6091  tfr1onlemaccex  6095  tfr1onlemres  6096  tfrcllemsucaccv  6101  tfrcllembxssdm  6103  tfrcllembfn  6104  tfrcllemaccex  6108  tfrcllemres  6109  tfrcl  6111  ordiso2  6707
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