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Theorem elon2 4403
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 4402 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 elex 2796 . . 3  |-  ( A  e.  On  ->  A  e.  _V )
31, 2jca 518 . 2  |-  ( A  e.  On  ->  ( Ord  A  /\  A  e. 
_V ) )
4 elong 4400 . . 3  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
54biimparc 473 . 2  |-  ( ( Ord  A  /\  A  e.  _V )  ->  A  e.  On )
63, 5impbii 180 1  |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   _Vcvv 2788   Ord word 4391   Oncon0 4392
This theorem is referenced by:  sucelon  4608  tfrlem12  6405  tfrlem13  6406  gruina  8440  nobndlem1  23757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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