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Theorem eloni 4267
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 4265 . 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 1465  Ord word 4254  Oncon0 4255
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260
This theorem is referenced by:  elon2  4268  onelon  4276  onin  4278  onelss  4279  ontr1  4281  onordi  4318  onss  4379  suceloni  4387  sucelon  4389  onsucmin  4393  onsucelsucr  4394  onintonm  4403  ordsucunielexmid  4416  onsucuni2  4449  nnord  4495  tfrlem1  6173  tfrlemisucaccv  6190  tfrlemibfn  6193  tfrlemiubacc  6195  tfrexlem  6199  tfr1onlemsucfn  6205  tfr1onlemsucaccv  6206  tfr1onlembfn  6209  tfr1onlemubacc  6211  tfrcllemsucfn  6218  tfrcllemsucaccv  6219  tfrcllembfn  6222  tfrcllemubacc  6224  sucinc2  6310  phplem4on  6729  ordiso  6889
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