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Theorem elon2 4419
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 4418 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 elex 2809 . . 3  |-  ( A  e.  On  ->  A  e.  _V )
31, 2jca 518 . 2  |-  ( A  e.  On  ->  ( Ord  A  /\  A  e. 
_V ) )
4 elong 4416 . . 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 1696   _Vcvv 2801   Ord word 4407   Oncon0 4408
This theorem is referenced by:  sucelon  4624  tfrlem12  6421  tfrlem13  6422  gruina  8456  nobndlem1  24417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844  df-tr 4130  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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