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Theorem oneli 6281
 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 6198 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
31, 2mpan 689 1 (𝐵𝐴𝐵 ∈ On)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  Oncon0 6173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-tr 5142  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-ord 6176  df-on 6177 This theorem is referenced by:  onssneli  6283  oawordeulem  8195  rankuni  9330  tcrank  9351  cardne  9432  cardval2  9458  alephsuc2  9545  cfsmolem  9735  cfcof  9739  alephreg  10047  pwcfsdom  10048  tskcard  10246  lrcut  33666  onsucconni  34201
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