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Theorem ordtr1 4626
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 4597 . 2  |-  ( Ord 
C  ->  Tr  C
)
2 trel 4311 . 2  |-  ( Tr  C  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )
31, 2syl 16 1  |-  ( Ord 
C  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   Tr wtr 4304   Ord word 4582
This theorem is referenced by:  ontr1  4629  dfsmo2  6611  smores2  6618  smoel  6624  smogt  6631  ordiso2  7486  r1ordg  7706  r1pwss  7712  r1val1  7714  rankr1ai  7726  rankval3b  7754  rankonidlem  7756  onssr1  7759  cofsmo  8151  fpwwe2lem9  8515  bnj1098  29216  bnj594  29345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-ss 3336  df-uni 4018  df-tr 4305  df-ord 4586
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