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Theorem elong 4400
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 4399 . 2  |-  ( x  =  A  ->  ( Ord  x  <->  Ord  A ) )
2 df-on 4396 . 2  |-  On  =  { x  |  Ord  x }
31, 2elab2g 2916 1  |-  ( A  e.  V  ->  ( A  e.  On  <->  Ord  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   Ord word 4391   Oncon0 4392
This theorem is referenced by:  elon  4401  eloni  4402  elon2  4403  ordelon  4416  onin  4423  limelon  4455  ordsssuc2  4481  onprc  4576  ssonuni  4578  suceloni  4604  ordsuc  4605  oion  7251  hartogs  7259  card2on  7268  tskwe  7583  onssnum  7667  hsmexlem1  8052  ondomon  8185  1stcrestlem  17178  hfninf  24816  celsor  25111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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