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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 33800 | . 2 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅}) | |
2 | difss 4022 | . . 3 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ (On ∩ Fix Bigcup ) | |
3 | inss1 4119 | . . 3 ⊢ (On ∩ Fix Bigcup ) ⊆ On | |
4 | 2, 3 | sstri 3886 | . 2 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ On |
5 | 1, 4 | eqsstri 3911 | 1 ⊢ Limits ⊆ On |
Colors of variables: wff setvar class |
Syntax hints: ∖ cdif 3840 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 {csn 4516 Oncon0 6172 Bigcup cbigcup 33774 Fix cfix 33775 Limits climits 33776 |
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 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 df-dif 3846 df-in 3850 df-ss 3860 df-limits 33800 |
This theorem is referenced by: (None) |
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