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Theorem elon 4896
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
Hypothesis
Ref Expression
elon.1  |-  A  e. 
_V
Assertion
Ref Expression
elon  |-  ( A  e.  On  <->  Ord  A )

Proof of Theorem elon
StepHypRef Expression
1 elon.1 . 2  |-  A  e. 
_V
2 elong 4895 . 2  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
31, 2ax-mp 5 1  |-  ( A  e.  On  <->  Ord  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1819   _Vcvv 3109   Ord word 4886   Oncon0 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3478  df-ss 3485  df-uni 4252  df-tr 4551  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891
This theorem is referenced by:  tron  4910  0elon  4940  smogt  7056  rdglim2  7116  omeulem1  7249  isfinite2  7796  r0weon  8407  cflim3  8659  inar1  9170  ellimits  29765  dford3lem2  31173
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