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Theorem kur14lem6 24889
 Description: Lemma for kur14 24894. If is the complementation operator and is the closure operator, this expresses the identity for any subset of the topological space. This is the key result that lets us cut down long enough sequences of that arise when applying closure and complement repeatedly to , and explains why we end up with a number as large as , yet no larger. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
kur14lem.j
kur14lem.x
kur14lem.k
kur14lem.i
kur14lem.a
kur14lem.b
Assertion
Ref Expression
kur14lem6

Proof of Theorem kur14lem6
StepHypRef Expression
1 kur14lem.j . . . . 5
2 kur14lem.x . . . . . 6
3 kur14lem.k . . . . . 6
4 kur14lem.i . . . . . 6
5 kur14lem.b . . . . . . 7
6 difss 3466 . . . . . . 7
75, 6eqsstri 3370 . . . . . 6
81, 2, 3, 4, 7kur14lem3 24886 . . . . 5
94fveq1i 5721 . . . . . 6
102ntrss2 17113 . . . . . . 7
111, 8, 10mp2an 654 . . . . . 6
129, 11eqsstri 3370 . . . . 5
132clsss 17110 . . . . 5
141, 8, 12, 13mp3an 1279 . . . 4
153fveq1i 5721 . . . 4
163fveq1i 5721 . . . 4
1714, 15, 163sstr4i 3379 . . 3
181, 2, 3, 4, 7kur14lem5 24888 . . 3
1917, 18sseqtri 3372 . 2
201, 2, 3, 4, 8kur14lem2 24885 . . . . 5
21 difss 3466 . . . . 5
2220, 21eqsstri 3370 . . . 4
23 kur14lem.a . . . . . . . . 9
241, 2, 3, 4, 23kur14lem3 24886 . . . . . . . 8
255fveq2i 5723 . . . . . . . . . . 11
2625difeq2i 3454 . . . . . . . . . 10
271, 2, 3, 4, 24kur14lem2 24885 . . . . . . . . . 10
284fveq1i 5721 . . . . . . . . . 10
2926, 27, 283eqtr2i 2461 . . . . . . . . 9
302ntrss2 17113 . . . . . . . . . 10
311, 24, 30mp2an 654 . . . . . . . . 9
3229, 31eqsstri 3370 . . . . . . . 8
332clsss 17110 . . . . . . . 8
341, 24, 32, 33mp3an 1279 . . . . . . 7
353fveq1i 5721 . . . . . . 7
361, 2, 3, 4, 23kur14lem5 24888 . . . . . . . 8
373fveq1i 5721 . . . . . . . 8
3836, 37eqtr3i 2457 . . . . . . 7
3934, 35, 383sstr4i 3379 . . . . . 6
40 sscon 3473 . . . . . 6
4139, 40ax-mp 8 . . . . 5
4241, 5, 203sstr4i 3379 . . . 4
432clsss 17110 . . . 4
441, 22, 42, 43mp3an 1279 . . 3
453fveq1i 5721 . . 3
4644, 45, 153sstr4i 3379 . 2
4719, 46eqssi 3356 1
 Colors of variables: wff set class Syntax hints:   wceq 1652   wcel 1725   cdif 3309   wss 3312  cuni 4007  cfv 5446  ctop 16950  cnt 17073  ccl 17074 This theorem is referenced by:  kur14lem7  24890 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-top 16955  df-cld 17075  df-ntr 17076  df-cls 17077
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