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Theorem ordtr1 4278
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 4268 . 2  |-  ( Ord 
C  ->  Tr  C
)
2 trel 4001 . 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 103    e. wcel 1463   Tr wtr 3994   Ord word 4252
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-uni 3705  df-tr 3995  df-iord 4256
This theorem is referenced by:  ontr1  4279  ordwe  4458  dfsmo2  6150  smores2  6157  smoel  6163  tfr1onlemsucaccv  6204  tfr1onlembxssdm  6206  tfr1onlembfn  6207  tfr1onlemaccex  6211  tfr1onlemres  6212  tfrcllemsucaccv  6217  tfrcllembxssdm  6219  tfrcllembfn  6220  tfrcllemaccex  6224  tfrcllemres  6225  tfrcl  6227  ordiso2  6886
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