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Theorem tron 4890
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 4536 . 2  |-  ( Tr  On  <->  A. x  e.  On  x  C_  On )
2 vex 3109 . . . . . . 7  |-  x  e. 
_V
32elon 4876 . . . . . 6  |-  ( x  e.  On  <->  Ord  x )
4 ordelord 4889 . . . . . 6  |-  ( ( Ord  x  /\  y  e.  x )  ->  Ord  y )
53, 4sylanb 470 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  Ord  y )
65ex 432 . . . 4  |-  ( x  e.  On  ->  (
y  e.  x  ->  Ord  y ) )
7 vex 3109 . . . . 5  |-  y  e. 
_V
87elon 4876 . . . 4  |-  ( y  e.  On  <->  Ord  y )
96, 8syl6ibr 227 . . 3  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  e.  On ) )
109ssrdv 3495 . 2  |-  ( x  e.  On  ->  x  C_  On )
111, 10mprgbir 2818 1  |-  Tr  On
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1823    C_ wss 3461   Tr wtr 4532   Ord word 4866   Oncon0 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871
This theorem is referenced by:  ordon  6591  onuninsuci  6648  gruina  9185
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