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Theorem elon2 5441
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 5440 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 elex 3040 . . 3  |-  ( A  e.  On  ->  A  e.  _V )
31, 2jca 541 . 2  |-  ( A  e.  On  ->  ( Ord  A  /\  A  e. 
_V ) )
4 elong 5438 . . 3  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
54biimparc 495 . 2  |-  ( ( Ord  A  /\  A  e.  _V )  ->  A  e.  On )
63, 5impbii 192 1  |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    e. wcel 1904   _Vcvv 3031   Ord word 5429   Oncon0 5430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-v 3033  df-in 3397  df-ss 3404  df-uni 4191  df-tr 4491  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-ord 5433  df-on 5434
This theorem is referenced by:  sucelon  6663  tfrlem12  7125  tfrlem13  7126  gruina  9261  nobndlem1  30652
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