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Theorem onelssi 4630
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 4564 . 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 1717    C_ wss 3263   Oncon0 4522
This theorem is referenced by:  onelini  4633  oneluni  4634  oawordeulem  6733  cardsdomelir  7793  carddom2  7797  cardaleph  7903  alephsing  8089  domtriomlem  8255  axdc3lem  8263  inar1  8583  nodenselem6  25364  nodense  25367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-v 2901  df-in 3270  df-ss 3277  df-uni 3958  df-tr 4244  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526
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