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Theorem onelssi 4682
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 4615 . 2  |-  ( A  e.  On  ->  ( B  e.  A  ->  B 
C_  A ) )
31, 2ax-mp 8 1  |-  ( B  e.  A  ->  B  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    C_ wss 3312   Oncon0 4573
This theorem is referenced by:  onelini  4685  oneluni  4686  oawordeulem  6789  cardsdomelir  7852  carddom2  7856  cardaleph  7962  alephsing  8148  domtriomlem  8314  axdc3lem  8322  inar1  8642  nodenselem6  25633  nodense  25636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-uni 4008  df-tr 4295  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
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