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Theorem onelssi 4986
Description: A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
Hypothesis
Ref Expression
on.1  |-  A  e.  On
Assertion
Ref Expression
onelssi  |-  ( B  e.  A  ->  B  C_  A )

Proof of Theorem onelssi
StepHypRef Expression
1 on.1 . 2  |-  A  e.  On
2 onelss 4920 . 2  |-  ( A  e.  On  ->  ( B  e.  A  ->  B 
C_  A ) )
31, 2ax-mp 5 1  |-  ( B  e.  A  ->  B  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767    C_ wss 3476   Oncon0 4878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-in 3483  df-ss 3490  df-uni 4246  df-tr 4541  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882
This theorem is referenced by:  onelini  4989  oneluni  4990  oawordeulem  7200  cardsdomelir  8350  carddom2  8354  cardaleph  8466  alephsing  8652  domtriomlem  8818  axdc3lem  8826  inar1  9149  nodenselem6  29023  nodense  29026
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