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Theorem ontr2 4438
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 4401 . 2  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4401 . 2  |-  ( C  e.  On  ->  Ord  C )
3 ordtr2 4435 . 2  |-  ( ( Ord  A  /\  Ord  C )  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  C ) )
41, 2, 3syl2an 465 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 6    /\ wa 360    e. wcel 1685    C_ wss 3153   Ord word 4390   Oncon0 4391
This theorem is referenced by:  oeordsuc  6587  oelimcl  6593  oeeui  6595  omopthlem2  6649  omxpenlem  6958  oismo  7250  cantnflem1c  7384  cantnflem1  7386  cantnflem3  7388  rankr1ai  7465  rankxplim  7544  infxpenlem  7636  alephle  7710  pwcfsdom  8200  r1limwun  8353  axfelem6  23752  ontopbas  24274  ontgval  24277
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395
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