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Theorem tron 4604
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 4306 . 2  |-  ( Tr  On  <->  A. x  e.  On  x  C_  On )
2 vex 2959 . . . . . . 7  |-  x  e. 
_V
32elon 4590 . . . . . 6  |-  ( x  e.  On  <->  Ord  x )
4 ordelord 4603 . . . . . 6  |-  ( ( Ord  x  /\  y  e.  x )  ->  Ord  y )
53, 4sylanb 459 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  Ord  y )
65ex 424 . . . 4  |-  ( x  e.  On  ->  (
y  e.  x  ->  Ord  y ) )
7 vex 2959 . . . . 5  |-  y  e. 
_V
87elon 4590 . . . 4  |-  ( y  e.  On  <->  Ord  y )
96, 8syl6ibr 219 . . 3  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  e.  On ) )
109ssrdv 3354 . 2  |-  ( x  e.  On  ->  x  C_  On )
111, 10mprgbir 2776 1  |-  Tr  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1725    C_ wss 3320   Tr wtr 4302   Ord word 4580   Oncon0 4581
This theorem is referenced by:  ordon  4763  onuninsuci  4820  gruina  8693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585
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