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Theorem elon2 4830
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 4829 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 elex 3065 . . 3  |-  ( A  e.  On  ->  A  e.  _V )
31, 2jca 530 . 2  |-  ( A  e.  On  ->  ( Ord  A  /\  A  e. 
_V ) )
4 elong 4827 . . 3  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
54biimparc 485 . 2  |-  ( ( Ord  A  /\  A  e.  _V )  ->  A  e.  On )
63, 5impbii 188 1  |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    e. wcel 1840   _Vcvv 3056   Ord word 4818   Oncon0 4819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ral 2756  df-rex 2757  df-v 3058  df-in 3418  df-ss 3425  df-uni 4189  df-tr 4487  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823
This theorem is referenced by:  sucelon  6588  tfrlem12  7013  tfrlem13  7014  gruina  9144  nobndlem1  30120
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