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Theorem ontr2 4455
Description: Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Nov-2003.)
Assertion
Ref Expression
ontr2  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( ( A  C_  B  /\  B  e.  C
)  ->  A  e.  C ) )

Proof of Theorem ontr2
StepHypRef Expression
1 eloni 4418 . 2  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4418 . 2  |-  ( C  e.  On  ->  Ord  C )
3 ordtr2 4452 . 2  |-  ( ( Ord  A  /\  Ord  C )  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  C ) )
41, 2, 3syl2an 463 1  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( ( A  C_  B  /\  B  e.  C
)  ->  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696    C_ wss 3165   Ord word 4407   Oncon0 4408
This theorem is referenced by:  oeordsuc  6608  oelimcl  6614  oeeui  6616  omopthlem2  6670  omxpenlem  6979  oismo  7271  cantnflem1c  7405  cantnflem1  7407  cantnflem3  7409  rankr1ai  7486  rankxplim  7565  infxpenlem  7657  alephle  7731  pwcfsdom  8221  r1limwun  8374  nobndlem6  24422  ontopbas  24939  ontgval  24942
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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