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Theorem oneli 5752
 Description: A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
Hypothesis
Ref Expression
on.1 𝐴 ∈ On
Assertion
Ref Expression
oneli (𝐵𝐴𝐵 ∈ On)

Proof of Theorem oneli
StepHypRef Expression
1 on.1 . 2 𝐴 ∈ On
2 onelon 5665 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
31, 2mpan 702 1 (𝐵𝐴𝐵 ∈ On)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644 This theorem is referenced by:  onssneli  5754  oawordeulem  7521  rankuni  8609  tcrank  8630  cardne  8674  cardval2  8700  alephsuc2  8786  cfsmolem  8975  cfcof  8979  alephreg  9283  pwcfsdom  9284  tskcard  9482  onsucconi  31606
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