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Theorem tron 4147
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 3886 . 2  |-  ( Tr  On  <->  A. x  e.  On  x  C_  On )
2 vex 2577 . . . . . . 7  |-  x  e. 
_V
32elon 4139 . . . . . 6  |-  ( x  e.  On  <->  Ord  x )
4 ordelord 4146 . . . . . 6  |-  ( ( Ord  x  /\  y  e.  x )  ->  Ord  y )
53, 4sylanb 272 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  Ord  y )
65ex 112 . . . 4  |-  ( x  e.  On  ->  (
y  e.  x  ->  Ord  y ) )
7 vex 2577 . . . . 5  |-  y  e. 
_V
87elon 4139 . . . 4  |-  ( y  e.  On  <->  Ord  y )
96, 8syl6ibr 155 . . 3  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  e.  On ) )
109ssrdv 2979 . 2  |-  ( x  e.  On  ->  x  C_  On )
111, 10mprgbir 2396 1  |-  Tr  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1409    C_ wss 2945   Tr wtr 3882   Ord word 4127   Oncon0 4128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133
This theorem is referenced by:  ordon  4240  tfi  4333
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