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Theorem oneli 5734
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 5647 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
31, 2mpan 701 1 (𝐵𝐴𝐵 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1975  Oncon0 5622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-tr 4671  df-eprel 4935  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-ord 5625  df-on 5626
This theorem is referenced by:  onssneli  5736  oawordeulem  7494  rankuni  8582  tcrank  8603  cardne  8647  cardval2  8673  alephsuc2  8759  cfsmolem  8948  cfcof  8952  alephreg  9256  pwcfsdom  9257  tskcard  9455  onsucconi  31408
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