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Theorem eloni 4265
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 4263 . 2  |-  ( A  e.  On  ->  ( A  e.  On  <->  Ord  A ) )
21ibi 175 1  |-  ( A  e.  On  ->  Ord  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    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:  elon2  4266  onelon  4274  onin  4276  onelss  4277  ontr1  4279  onordi  4316  onss  4377  suceloni  4385  sucelon  4387  onsucmin  4391  onsucelsucr  4392  onintonm  4401  ordsucunielexmid  4414  onsucuni2  4447  nnord  4493  tfrlem1  6171  tfrlemisucaccv  6188  tfrlemibfn  6191  tfrlemiubacc  6193  tfrexlem  6197  tfr1onlemsucfn  6203  tfr1onlemsucaccv  6204  tfr1onlembfn  6207  tfr1onlemubacc  6209  tfrcllemsucfn  6216  tfrcllemsucaccv  6217  tfrcllembfn  6220  tfrcllemubacc  6222  sucinc2  6308  phplem4on  6727  ordiso  6887
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