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Theorem ontr2 4411
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 4374 . 2  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4374 . 2  |-  ( C  e.  On  ->  Ord  C )
3 ordtr2 4408 . 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 1621    C_ wss 3127   Ord word 4363   Oncon0 4364
This theorem is referenced by:  oeordsuc  6560  oelimcl  6566  oeeui  6568  omopthlem2  6622  omxpenlem  6931  oismo  7223  cantnflem1c  7357  cantnflem1  7359  cantnflem3  7361  rankr1ai  7438  rankxplim  7517  infxpenlem  7609  alephle  7683  pwcfsdom  8173  r1limwun  8326  axfelem6  23720  ontopbas  24242  ontgval  24245
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368
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