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Theorem ordtr1 4927
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 4898 . 2  |-  ( Ord 
C  ->  Tr  C
)
2 trel 4553 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   Tr wtr 4546   Ord word 4883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-in 3488  df-ss 3495  df-uni 4252  df-tr 4547  df-ord 4887
This theorem is referenced by:  ontr1  4930  dfsmo2  7030  smores2  7037  smoel  7043  smogt  7050  ordiso2  7952  r1ordg  8208  r1pwss  8214  r1val1  8216  rankr1ai  8228  rankval3b  8256  rankonidlem  8258  onssr1  8261  cofsmo  8661  fpwwe2lem9  9028  bnj1098  33322  bnj594  33450
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