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Theorem eloni 4202
Description: An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
Assertion
Ref Expression
eloni  |-  ( A  e.  On  ->  Ord  A )

Proof of Theorem eloni
StepHypRef Expression
1 elong 4200 . 2  |-  ( A  e.  On  ->  ( A  e.  On  <->  Ord  A ) )
21ibi 174 1  |-  ( A  e.  On  ->  Ord  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   Ord word 4189   Oncon0 4190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195
This theorem is referenced by:  elon2  4203  onelon  4211  onin  4213  onelss  4214  ontr1  4216  onordi  4253  onss  4310  suceloni  4318  sucelon  4320  onsucmin  4324  onsucelsucr  4325  onintonm  4334  ordsucunielexmid  4347  onsucuni2  4380  nnord  4426  tfrlem1  6073  tfrlemisucaccv  6090  tfrlemibfn  6093  tfrlemiubacc  6095  tfrexlem  6099  tfr1onlemsucfn  6105  tfr1onlemsucaccv  6106  tfr1onlembfn  6109  tfr1onlemubacc  6111  tfrcllemsucfn  6118  tfrcllemsucaccv  6119  tfrcllembfn  6122  tfrcllemubacc  6124  sucinc2  6207  phplem4on  6583  ordiso  6729
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