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Theorem onelssi 5824
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 5754 . 2 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
31, 2ax-mp 5 1 (𝐵𝐴𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988  wss 3567  Oncon0 5711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-in 3574  df-ss 3581  df-uni 4428  df-tr 4744  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-ord 5714  df-on 5715
This theorem is referenced by:  onelini  5827  oneluni  5828  oawordeulem  7619  cardsdomelir  8784  carddom2  8788  cardaleph  8897  alephsing  9083  domtriomlem  9249  axdc3lem  9257  inar1  9582  nodense  31816
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