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Theorem tron 4845
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 4492 . 2  |-  ( Tr  On  <->  A. x  e.  On  x  C_  On )
2 vex 3075 . . . . . . 7  |-  x  e. 
_V
32elon 4831 . . . . . 6  |-  ( x  e.  On  <->  Ord  x )
4 ordelord 4844 . . . . . 6  |-  ( ( Ord  x  /\  y  e.  x )  ->  Ord  y )
53, 4sylanb 472 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  Ord  y )
65ex 434 . . . 4  |-  ( x  e.  On  ->  (
y  e.  x  ->  Ord  y ) )
7 vex 3075 . . . . 5  |-  y  e. 
_V
87elon 4831 . . . 4  |-  ( y  e.  On  <->  Ord  y )
96, 8syl6ibr 227 . . 3  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  e.  On ) )
109ssrdv 3465 . 2  |-  ( x  e.  On  ->  x  C_  On )
111, 10mprgbir 2898 1  |-  Tr  On
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758    C_ wss 3431   Tr wtr 4488   Ord word 4821   Oncon0 4822
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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-tr 4489  df-eprel 4735  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826
This theorem is referenced by:  ordon  6499  onuninsuci  6556  gruina  9091
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