Theorem limitssson 34140

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 34089	. 2 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
2		difss 4062	. . 3 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ (On ∩ Fix Bigcup )
3		inss1 4159	. . 3 ⊢ (On ∩ Fix Bigcup ) ⊆ On
4	2, 3	sstri 3926	. 2 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3951	1 ⊢ Limits ⊆ On

Syntax hints:   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558  Oncon0 6251   Bigcup cbigcup 34063   Fix cfix 34064   Limits climits 34065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-limits 34089

This theorem is referenced by: (None)