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Theorem elon 4137
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
Hypothesis
Ref Expression
elon.1  |-  A  e. 
_V
Assertion
Ref Expression
elon  |-  ( A  e.  On  <->  Ord  A )

Proof of Theorem elon
StepHypRef Expression
1 elon.1 . 2  |-  A  e. 
_V
2 elong 4136 . 2  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
31, 2ax-mp 7 1  |-  ( A  e.  On  <->  Ord  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1434   _Vcvv 2602   Ord word 4125   Oncon0 4126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-in 2980  df-ss 2987  df-uni 3610  df-tr 3884  df-iord 4129  df-on 4131
This theorem is referenced by:  tron  4145  0elon  4155  ordtriexmidlem  4271  ontr2exmid  4276  ordtri2or2exmidlem  4277  onsucelsucexmidlem  4280  bj-omelon  10941
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