Theorem limitssson 35875

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 35824	. 2 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
2		difss 4159	. . 3 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ (On ∩ Fix Bigcup )
3		inss1 4258	. . 3 ⊢ (On ∩ Fix Bigcup ) ⊆ On
4	2, 3	sstri 4018	. 2 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ On
5	1, 4	eqsstri 4043	1 ⊢ Limits ⊆ On

Syntax hints:   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648  Oncon0 6395   Bigcup cbigcup 35798   Fix cfix 35799   Limits climits 35800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-in 3983  df-ss 3993  df-limits 35824

This theorem is referenced by: (None)