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Theorem elong 4828
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 4827 . 2  |-  ( x  =  A  ->  ( Ord  x  <->  Ord  A ) )
2 df-on 4824 . 2  |-  On  =  { x  |  Ord  x }
31, 2elab2g 3208 1  |-  ( A  e.  V  ->  ( A  e.  On  <->  Ord  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1758   Ord word 4819   Oncon0 4820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-v 3073  df-in 3436  df-ss 3443  df-uni 4193  df-tr 4487  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824
This theorem is referenced by:  elon  4829  eloni  4830  elon2  4831  ordelon  4844  onin  4851  limelon  4883  ordsssuc2  4908  onprc  6499  ssonuni  6501  suceloni  6527  ordsuc  6528  oion  7854  hartogs  7862  card2on  7873  tskwe  8224  onssnum  8314  hsmexlem1  8699  ondomon  8831  1stcrestlem  19181  hfninf  28361
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