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Theorem tron 4452
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 4154 . 2  |-  ( Tr  On  <->  A. x  e.  On  x  C_  On )
2 vex 2825 . . . . . . 7  |-  x  e. 
_V
32elon 4438 . . . . . 6  |-  ( x  e.  On  <->  Ord  x )
4 ordelord 4451 . . . . . 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 2825 . . . . 5  |-  y  e. 
_V
87elon 4438 . . . 4  |-  ( y  e.  On  <->  Ord  y )
96, 8syl6ibr 218 . . 3  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  e.  On ) )
109ssrdv 3219 . 2  |-  ( x  e.  On  ->  x  C_  On )
111, 10mprgbir 2647 1  |-  Tr  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1701    C_ wss 3186   Tr wtr 4150   Ord word 4428   Oncon0 4429
This theorem is referenced by:  ordon  4611  onuninsuci  4668  gruina  8485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433
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