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Theorem limitssson 34213
Description: The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
Assertion
Ref Expression
limitssson Limits ⊆ On

Proof of Theorem limitssson
StepHypRef Expression
1 df-limits 34162 . 2 Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
2 difss 4066 . . 3 ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ (On ∩ Fix Bigcup )
3 inss1 4162 . . 3 (On ∩ Fix Bigcup ) ⊆ On
42, 3sstri 3930 . 2 ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ On
51, 4eqsstri 3955 1 Limits ⊆ On
Colors of variables: wff setvar class
Syntax hints:  cdif 3884  cin 3886  wss 3887  c0 4256  {csn 4561  Oncon0 6266   Bigcup cbigcup 34136   Fix cfix 34137   Limits climits 34138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-limits 34162
This theorem is referenced by: (None)
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