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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem4 | Structured version Visualization version GIF version |
Description: Lemma for kur14 34024. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
kur14lem.j | β’ π½ β Top |
kur14lem.x | β’ π = βͺ π½ |
kur14lem.k | β’ πΎ = (clsβπ½) |
kur14lem.i | β’ πΌ = (intβπ½) |
kur14lem.a | β’ π΄ β π |
Ref | Expression |
---|---|
kur14lem4 | β’ (π β (π β π΄)) = π΄ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kur14lem.a | . 2 β’ π΄ β π | |
2 | dfss4 4253 | . 2 β’ (π΄ β π β (π β (π β π΄)) = π΄) | |
3 | 1, 2 | mpbi 229 | 1 β’ (π β (π β π΄)) = π΄ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 β wcel 2106 β cdif 3940 β wss 3943 βͺ cuni 4900 βcfv 6531 Topctop 22321 intcnt 22447 clsccl 22448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3474 df-dif 3946 df-in 3950 df-ss 3960 |
This theorem is referenced by: kur14lem7 34020 |
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