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Theorem eloni 4213
Description: An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
Assertion
Ref Expression
eloni (𝐴 ∈ On → Ord 𝐴)

Proof of Theorem eloni
StepHypRef Expression
1 elong 4211 . 2 (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
21ibi 175 1 (𝐴 ∈ On → Ord 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1439  Ord word 4200  Oncon0 4201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-in 3008  df-ss 3015  df-uni 3662  df-tr 3945  df-iord 4204  df-on 4206
This theorem is referenced by:  elon2  4214  onelon  4222  onin  4224  onelss  4225  ontr1  4227  onordi  4264  onss  4325  suceloni  4333  sucelon  4335  onsucmin  4339  onsucelsucr  4340  onintonm  4349  ordsucunielexmid  4362  onsucuni2  4395  nnord  4441  tfrlem1  6089  tfrlemisucaccv  6106  tfrlemibfn  6109  tfrlemiubacc  6111  tfrexlem  6115  tfr1onlemsucfn  6121  tfr1onlemsucaccv  6122  tfr1onlembfn  6125  tfr1onlemubacc  6127  tfrcllemsucfn  6134  tfrcllemsucaccv  6135  tfrcllembfn  6138  tfrcllemubacc  6140  sucinc2  6223  phplem4on  6639  ordiso  6785
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