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Theorem isclo2 20181
 Description: A set is clopen iff for every point in the space there is a neighborhood of which is either disjoint from or contained in . (Contributed by Mario Carneiro, 7-Jul-2015.)
Hypothesis
Ref Expression
isclo.1
Assertion
Ref Expression
isclo2
Distinct variable groups:   ,,,   ,,,   ,,,

Proof of Theorem isclo2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 isclo.1 . . 3
21isclo 20180 . 2
3 eleq1 2537 . . . . . . . . . . 11
43bibi2d 325 . . . . . . . . . 10
54cbvralv 3005 . . . . . . . . 9
65anbi2i 708 . . . . . . . 8
7 pm4.24 655 . . . . . . . 8
8 raaanv 3869 . . . . . . . 8
96, 7, 83bitr4i 285 . . . . . . 7
10 bibi1 334 . . . . . . . . . . . . 13
1110biimpa 492 . . . . . . . . . . . 12
1211biimpcd 232 . . . . . . . . . . 11
1312ralimdv 2806 . . . . . . . . . 10
1413com12 31 . . . . . . . . 9
15 dfss3 3408 . . . . . . . . 9
1614, 15syl6ibr 235 . . . . . . . 8
1716ralimi 2796 . . . . . . 7
189, 17sylbi 200 . . . . . 6
19 eleq1 2537 . . . . . . . . . . 11
2019imbi1d 324 . . . . . . . . . 10
2120rspcv 3132 . . . . . . . . 9
22 dfss3 3408 . . . . . . . . . . 11
2322imbi2i 319 . . . . . . . . . 10
24 r19.21v 2803 . . . . . . . . . 10
2523, 24bitr4i 260 . . . . . . . . 9
2621, 25syl6ib 234 . . . . . . . 8
27 ssel 3412 . . . . . . . . . . 11
2827com12 31 . . . . . . . . . 10
2928imim2d 53 . . . . . . . . 9
3029ralimdv 2806 . . . . . . . 8
3126, 30jcad 542 . . . . . . 7
32 ralbiim 2909 . . . . . . 7
3331, 32syl6ibr 235 . . . . . 6
3418, 33impbid2 209 . . . . 5
3534pm5.32i 649 . . . 4
3635rexbii 2881 . . 3
3736ralbii 2823 . 2
382, 37syl6bb 269 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   wceq 1452   wcel 1904  wral 2756  wrex 2757   cin 3389   wss 3390  cuni 4190  cfv 5589  ctop 19994  ccld 20108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-topgen 15420  df-top 19998  df-cld 20111 This theorem is referenced by:  conpcon  30030
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