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Theorem onelssi 4517
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 4450 . 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 1696    C_ wss 3165   Oncon0 4408
This theorem is referenced by:  onelini  4520  oneluni  4521  oawordeulem  6568  cardsdomelir  7622  carddom2  7626  cardaleph  7732  alephsing  7918  domtriomlem  8084  axdc3lem  8092  inar1  8413  nodenselem6  24411  nodense  24414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844  df-tr 4130  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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