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Theorem elong 4437
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
Assertion
Ref Expression
elong  |-  ( A  e.  V  ->  ( A  e.  On  <->  Ord  A ) )

Proof of Theorem elong
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ordeq 4436 . 2  |-  ( x  =  A  ->  ( Ord  x  <->  Ord  A ) )
2 df-on 4433 . 2  |-  On  =  { x  |  Ord  x }
31, 2elab2g 2950 1  |-  ( A  e.  V  ->  ( A  e.  On  <->  Ord  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1701   Ord word 4428   Oncon0 4429
This theorem is referenced by:  elon  4438  eloni  4439  elon2  4440  ordelon  4453  onin  4460  limelon  4492  ordsssuc2  4518  onprc  4613  ssonuni  4615  suceloni  4641  ordsuc  4642  oion  7296  hartogs  7304  card2on  7313  tskwe  7628  onssnum  7712  hsmexlem1  8097  ondomon  8230  1stcrestlem  17234  hfninf  25202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-v 2824  df-in 3193  df-ss 3200  df-uni 3865  df-tr 4151  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433
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