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Mirrors > Home > MPE Home > Th. List > Mathboxes > limitssson | Structured version Visualization version GIF version |
Description: The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
limitssson | ⊢ Limits ⊆ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-limits 34162 | . 2 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅}) | |
2 | difss 4066 | . . 3 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ (On ∩ Fix Bigcup ) | |
3 | inss1 4162 | . . 3 ⊢ (On ∩ Fix Bigcup ) ⊆ On | |
4 | 2, 3 | sstri 3930 | . 2 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ On |
5 | 1, 4 | eqsstri 3955 | 1 ⊢ Limits ⊆ On |
Colors of variables: wff setvar class |
Syntax hints: ∖ cdif 3884 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 {csn 4561 Oncon0 6266 Bigcup cbigcup 34136 Fix cfix 34137 Limits climits 34138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3434 df-dif 3890 df-in 3894 df-ss 3904 df-limits 34162 |
This theorem is referenced by: (None) |
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