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Theorem onelssi 6501
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 6428 . 2 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
31, 2ax-mp 5 1 (𝐵𝐴𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wss 3963  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
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-ss 3980  df-uni 4913  df-tr 5266  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  onelini  6504  oneluni  6505  oawordeulem  8591  cardsdomelir  10011  carddom2  10015  cardaleph  10127  alephsing  10314  domtriomlem  10480  axdc3lem  10488  inar1  10813  nodense  27752
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