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Theorem eloni 4292
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 4290 . 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 1480  Ord word 4279  Oncon0 4280
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285
This theorem is referenced by:  elon2  4293  onelon  4301  onin  4303  onelss  4304  ontr1  4306  onordi  4343  onss  4404  suceloni  4412  sucelon  4414  onsucmin  4418  onsucelsucr  4419  onintonm  4428  ordsucunielexmid  4441  onsucuni2  4474  nnord  4520  tfrlem1  6198  tfrlemisucaccv  6215  tfrlemibfn  6218  tfrlemiubacc  6220  tfrexlem  6224  tfr1onlemsucfn  6230  tfr1onlemsucaccv  6231  tfr1onlembfn  6234  tfr1onlemubacc  6236  tfrcllemsucfn  6243  tfrcllemsucaccv  6244  tfrcllembfn  6247  tfrcllemubacc  6249  sucinc2  6335  phplem4on  6754  ordiso  6914
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