Theorem limitssson 36490

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 36439	. 2 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
2		difss 4086	. . 3 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ (On ∩ Fix Bigcup )
3		inss1 4185	. . 3 ⊢ (On ∩ Fix Bigcup ) ⊆ On
4	2, 3	sstri 3943	. 2 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3980	1 ⊢ Limits ⊆ On

This proof depends on syntax axioms:   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  {csn 4587  Oncon0 6361   Bigcup cbigcup 36413   Fix cfix 36414   Limits climits 36415

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-dif 3905  df-in 3909  df-ss 3919  df-limits 36439

This theorem is used by: (None)