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Theorem eloni 4348
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 4346 . 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 2135  Ord word 4335  Oncon0 4336
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-in 3118  df-ss 3125  df-uni 3785  df-tr 4076  df-iord 4339  df-on 4341
This theorem is referenced by:  elon2  4349  onelon  4357  onin  4359  onelss  4360  ontr1  4362  onordi  4399  onss  4465  suceloni  4473  sucelon  4475  onsucmin  4479  onsucelsucr  4480  onintonm  4489  ordsucunielexmid  4503  onsucuni2  4536  nnord  4584  tfrlem1  6268  tfrlemisucaccv  6285  tfrlemibfn  6288  tfrlemiubacc  6290  tfrexlem  6294  tfr1onlemsucfn  6300  tfr1onlemsucaccv  6301  tfr1onlembfn  6304  tfr1onlemubacc  6306  tfrcllemsucfn  6313  tfrcllemsucaccv  6314  tfrcllembfn  6317  tfrcllemubacc  6319  sucinc2  6406  phplem4on  6825  ordiso  6993
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