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Theorem onelon 4175
Description: An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
onelon ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)

Proof of Theorem onelon
StepHypRef Expression
1 eloni 4166 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordelon 4174 . 2 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
31, 2sylan 277 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1434  Ord word 4153  Oncon0 4154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-in 2990  df-ss 2997  df-uni 3628  df-tr 3902  df-iord 4157  df-on 4159
This theorem is referenced by:  oneli  4219  ssorduni  4267  unon  4291  tfrlemibacc  6023  tfrlemibxssdm  6024  tfrlemibfn  6025  tfrexlem  6031  tfr1onlemsucaccv  6038  tfrcllemsucaccv  6051  sucinc2  6139  oav2  6156  omv2  6158
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