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Theorem onelon 4121
 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 4112 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordelon 4120 . 2 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
31, 2sylan 267 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393  Ord word 4099  Oncon0 4100 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105 This theorem is referenced by:  oneli  4165  ssorduni  4213  unon  4237  tfrlemibacc  5940  tfrlemibxssdm  5941  tfrlemibfn  5942  tfrexlem  5948  sucinc2  6026  oav2  6043  omv2  6045
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