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Theorem elong 4879
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 4878 . 2  |-  ( x  =  A  ->  ( Ord  x  <->  Ord  A ) )
2 df-on 4875 . 2  |-  On  =  { x  |  Ord  x }
31, 2elab2g 3245 1  |-  ( A  e.  V  ->  ( A  e.  On  <->  Ord  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1762   Ord word 4870   Oncon0 4871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-v 3108  df-in 3476  df-ss 3483  df-uni 4239  df-tr 4534  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875
This theorem is referenced by:  elon  4880  eloni  4881  elon2  4882  ordelon  4895  onin  4902  limelon  4934  ordsssuc2  4959  onprc  6591  ssonuni  6593  suceloni  6619  ordsuc  6620  oion  7950  hartogs  7958  card2on  7969  tskwe  8320  onssnum  8410  hsmexlem1  8795  ondomon  8927  1stcrestlem  19712  hfninf  29406
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