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Theorem oneli 6500
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 6411 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
31, 2mpan 690 1 (𝐵𝐴𝐵 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  onssneli  6502  oawordeulem  8591  rankuni  9901  tcrank  9922  cardne  10003  cardval2  10029  alephsuc2  10118  cfsmolem  10308  cfcof  10312  alephreg  10620  pwcfsdom  10621  tskcard  10819  lrcut  27956  onsucconni  36420
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