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Theorem onelssi 6282
 Description: A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
Hypothesis
Ref Expression
on.1 𝐴 ∈ On
Assertion
Ref Expression
onelssi (𝐵𝐴𝐵𝐴)

Proof of Theorem onelssi
StepHypRef Expression
1 on.1 . 2 𝐴 ∈ On
2 onelss 6215 . 2 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
31, 2ax-mp 5 1 (𝐵𝐴𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111   ⊆ wss 3860  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-11 2158  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-v 3411  df-in 3867  df-ss 3877  df-uni 4802  df-tr 5142  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-ord 6176  df-on 6177 This theorem is referenced by:  onelini  6285  oneluni  6286  oawordeulem  8195  cardsdomelir  9440  carddom2  9444  cardaleph  9554  alephsing  9741  domtriomlem  9907  axdc3lem  9915  inar1  10240  nodense  33484
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