Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
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 31136 | . 2 ⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅}) | |
2 | difss 3699 | . . 3 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ (On ∩ Fix Bigcup ) | |
3 | inss1 3795 | . . 3 ⊢ (On ∩ Fix Bigcup ) ⊆ On | |
4 | 2, 3 | sstri 3577 | . 2 ⊢ ((On ∩ Fix Bigcup ) ∖ {∅}) ⊆ On |
5 | 1, 4 | eqsstri 3598 | 1 ⊢ Limits ⊆ On |
Colors of variables: wff setvar class |
Syntax hints: ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 Oncon0 5640 Bigcup cbigcup 31110 Fix cfix 31111 Limits climits 31112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-limits 31136 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |