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Theorem onelon 4316
 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 4307 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordelon 4315 . 2 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
31, 2sylan 281 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1481  Ord word 4294  Oncon0 4295 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-in 3083  df-ss 3090  df-uni 3746  df-tr 4036  df-iord 4298  df-on 4300 This theorem is referenced by:  oneli  4360  ssorduni  4413  unon  4437  tfrlemibacc  6234  tfrlemibxssdm  6235  tfrlemibfn  6236  tfrexlem  6242  tfr1onlemsucaccv  6249  tfrcllemsucaccv  6262  sucinc2  6353  oav2  6370  omv2  6372
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