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Theorem tron 4415
Description: The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
Assertion
Ref Expression
tron  |-  Tr  On

Proof of Theorem tron
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr3 4117 . 2  |-  ( Tr  On  <->  A. x  e.  On  x  C_  On )
2 vex 2791 . . . . . . 7  |-  x  e. 
_V
32elon 4401 . . . . . 6  |-  ( x  e.  On  <->  Ord  x )
4 ordelord 4414 . . . . . 6  |-  ( ( Ord  x  /\  y  e.  x )  ->  Ord  y )
53, 4sylanb 458 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  Ord  y )
65ex 423 . . . 4  |-  ( x  e.  On  ->  (
y  e.  x  ->  Ord  y ) )
7 vex 2791 . . . . 5  |-  y  e. 
_V
87elon 4401 . . . 4  |-  ( y  e.  On  <->  Ord  y )
96, 8syl6ibr 218 . . 3  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  e.  On ) )
109ssrdv 3185 . 2  |-  ( x  e.  On  ->  x  C_  On )
111, 10mprgbir 2613 1  |-  Tr  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1684    C_ wss 3152   Tr wtr 4113   Ord word 4391   Oncon0 4392
This theorem is referenced by:  ordon  4574  onuninsuci  4631  gruina  8440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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