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Theorem elon2 4203
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 4202 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 elex 2630 . . 3  |-  ( A  e.  On  ->  A  e.  _V )
31, 2jca 300 . 2  |-  ( A  e.  On  ->  ( Ord  A  /\  A  e. 
_V ) )
4 elong 4200 . . 3  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
54biimparc 293 . 2  |-  ( ( Ord  A  /\  A  e.  _V )  ->  A  e.  On )
63, 5impbii 124 1  |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    e. wcel 1438   _Vcvv 2619   Ord word 4189   Oncon0 4190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195
This theorem is referenced by:  tfrexlem  6099
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