Theorem limitssson 36137

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 36086	. 2 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
2		difss 4066	. . 3 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ (On ∩ Fix Bigcup )
3		inss1 4165	. . 3 ⊢ (On ∩ Fix Bigcup ) ⊆ On
4	2, 3	sstri 3924	. 2 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3961	1 ⊢ Limits ⊆ On

Syntax hints:   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  {csn 4555  Oncon0 6310   Bigcup cbigcup 36060   Fix cfix 36061   Limits climits 36062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-dif 3886  df-in 3890  df-ss 3900  df-limits 36086

This theorem is referenced by: (None)