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Theorem elong 4263
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 4262 . 2  |-  ( x  =  A  ->  ( Ord  x  <->  Ord  A ) )
2 df-on 4258 . 2  |-  On  =  { x  |  Ord  x }
31, 2elab2g 2802 1  |-  ( A  e.  V  ->  ( A  e.  On  <->  Ord  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1463   Ord word 4252   Oncon0 4253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-in 3045  df-ss 3052  df-uni 3705  df-tr 3995  df-iord 4256  df-on 4258
This theorem is referenced by:  elon  4264  eloni  4265  elon2  4266  ordelon  4273  onin  4276  limelon  4289  ssonuni  4372  suceloni  4385  sucelon  4387  onintonm  4401  onprc  4435  omelon2  4489  bj-nnelon  12959
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