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Theorem eloni 4140
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 4138 . 2 (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴))
21ibi 169 1 (𝐴 ∈ On → Ord 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1409  Ord word 4127  Oncon0 4128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133
This theorem is referenced by:  elon2  4141  onelon  4149  onin  4151  onelss  4152  ontr1  4154  onordi  4191  onss  4247  suceloni  4255  sucelon  4257  onsucmin  4261  onsucelsucr  4262  onintonm  4271  ordsucunielexmid  4284  onsucuni2  4316  nnord  4362  tfrlem1  5954  tfrlemisucaccv  5970  tfrlemibfn  5973  tfrlemiubacc  5975  tfrexlem  5979  sucinc2  6057  phplem4on  6360  ordiso  6416
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