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Theorem ordtr1 4451
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 4422 . 2  |-  ( Ord 
C  ->  Tr  C
)
2 trel 4136 . 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 1696   Tr wtr 4129   Ord word 4407
This theorem is referenced by:  ontr1  4454  dfsmo2  6380  smores2  6387  smoel  6393  smogt  6400  ordiso2  7246  r1ordg  7466  r1pwss  7472  r1val1  7474  rankr1ai  7486  rankval3b  7514  rankonidlem  7516  onssr1  7519  cofsmo  7911  fpwwe2lem9  8276  tartarmap  25991  bnj1098  29131  bnj594  29260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844  df-tr 4130  df-ord 4411
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