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Theorem onelssi 4900
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 4834 . 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 1826    C_ wss 3389   Oncon0 4792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rex 2738  df-v 3036  df-in 3396  df-ss 3403  df-uni 4164  df-tr 4461  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796
This theorem is referenced by:  onelini  4903  oneluni  4904  oawordeulem  7121  cardsdomelir  8267  carddom2  8271  cardaleph  8383  alephsing  8569  domtriomlem  8735  axdc3lem  8743  inar1  9064  nodenselem6  29611  nodense  29614
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