Theorem limitssson 33367

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

Step	Hyp	Ref	Expression
1		df-limits 33316	. 2 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
2		difss 4107	. . 3 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ (On ∩ Fix Bigcup )
3		inss1 4204	. . 3 ⊢ (On ∩ Fix Bigcup ) ⊆ On
4	2, 3	sstri 3975	. 2 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ On
5	1, 4	eqsstri 4000	1 ⊢ Limits ⊆ On

Syntax hints:   ∖ cdif 3932   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  {csn 4560  Oncon0 6185   Bigcup cbigcup 33290   Fix cfix 33291   Limits climits 33292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3938  df-in 3942  df-ss 3951  df-limits 33316

This theorem is referenced by: (None)