MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ontr2 Structured version   Unicode version

Theorem ontr2 4866
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 4829 . 2  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4829 . 2  |-  ( C  e.  On  ->  Ord  C )
3 ordtr2 4863 . 2  |-  ( ( Ord  A  /\  Ord  C )  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  C ) )
41, 2, 3syl2an 477 1  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( ( A  C_  B  /\  B  e.  C
)  ->  A  e.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758    C_ wss 3428   Ord word 4818   Oncon0 4819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-tr 4486  df-eprel 4732  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823
This theorem is referenced by:  oeordsuc  7135  oelimcl  7141  oeeui  7143  omopthlem2  7197  omxpenlem  7514  oismo  7857  cantnflem1c  7998  cantnflem1  8000  cantnflem3  8002  cantnflem1cOLD  8021  cantnflem1OLD  8023  cantnflem3OLD  8024  rankr1ai  8108  rankxplim  8189  infxpenlem  8283  alephle  8361  pwcfsdom  8850  r1limwun  9006  nobndlem6  27974  ontopbas  28410  ontgval  28413
  Copyright terms: Public domain W3C validator