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Theorem elon2 4621
Description: An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
Assertion
Ref Expression
elon2  |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )

Proof of Theorem elon2
StepHypRef Expression
1 eloni 4620 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 elex 2970 . . 3  |-  ( A  e.  On  ->  A  e.  _V )
31, 2jca 520 . 2  |-  ( A  e.  On  ->  ( Ord  A  /\  A  e. 
_V ) )
4 elong 4618 . . 3  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
54biimparc 475 . 2  |-  ( ( Ord  A  /\  A  e.  _V )  ->  A  e.  On )
63, 5impbii 182 1  |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    e. wcel 1727   _Vcvv 2962   Ord word 4609   Oncon0 4610
This theorem is referenced by:  sucelon  4826  tfrlem12  6679  tfrlem13  6680  gruina  8724  nobndlem1  25678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-v 2964  df-in 3313  df-ss 3320  df-uni 4040  df-tr 4328  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614
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