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Theorem elon 4395
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 4394 . 2  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
31, 2ax-mp 5 1  |-  ( A  e.  On  <->  Ord  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    e. wcel 2160   _Vcvv 2752   Ord word 4383   Oncon0 4384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-uni 3828  df-tr 4120  df-iord 4387  df-on 4389
This theorem is referenced by:  tron  4403  0elon  4413  ordtriexmidlem  4539  ontr2exmid  4545  ordtri2or2exmidlem  4546  onsucelsucexmidlem  4549  exmidonfinlem  7227  bj-omelon  15199
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