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Theorem onelon 4402
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 4393 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordelon 4401 . 2 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
31, 2sylan 283 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2160  Ord word 4380  Oncon0 4381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825  df-tr 4117  df-iord 4384  df-on 4386
This theorem is referenced by:  oneli  4446  ssorduni  4504  unon  4528  tfrlemibacc  6352  tfrlemibxssdm  6353  tfrlemibfn  6354  tfrexlem  6360  tfr1onlemsucaccv  6367  tfrcllemsucaccv  6380  sucinc2  6472  oav2  6489  omv2  6491
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