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Theorem ordtr1 4435
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 4406 . 2  |-  ( Ord 
C  ->  Tr  C
)
2 trel 4120 . 2  |-  ( Tr  C  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )
31, 2syl 15 1  |-  ( Ord 
C  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   Tr wtr 4113   Ord word 4391
This theorem is referenced by:  ontr1  4438  dfsmo2  6364  smores2  6371  smoel  6377  smogt  6384  ordiso2  7230  r1ordg  7450  r1pwss  7456  r1val1  7458  rankr1ai  7470  rankval3b  7498  rankonidlem  7500  onssr1  7503  cofsmo  7895  fpwwe2lem9  8260  tartarmap  25888  bnj1098  28815  bnj594  28944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114  df-ord 4395
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