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Theorem limitssson 33275
 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 33224 . 2 Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
2 difss 4112 . . 3 ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ (On ∩ Fix Bigcup )
3 inss1 4209 . . 3 (On ∩ Fix Bigcup ) ⊆ On
42, 3sstri 3980 . 2 ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ On
51, 4eqsstri 4005 1 Limits ⊆ On
 Colors of variables: wff setvar class Syntax hints:   ∖ cdif 3937   ∩ cin 3939   ⊆ wss 3940  ∅c0 4295  {csn 4564  Oncon0 6190   Bigcup cbigcup 33198   Fix cfix 33199   Limits climits 33200 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-v 3502  df-dif 3943  df-in 3947  df-ss 3956  df-limits 33224 This theorem is referenced by: (None)
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