Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  isclo2 Structured version   Unicode version

Theorem isclo2 17154
 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 17153 . 2
3 eleq1 2498 . . . . . . . . . . 11
43bibi2d 311 . . . . . . . . . 10
54cbvralv 2934 . . . . . . . . 9
65anbi2i 677 . . . . . . . 8
7 pm4.24 626 . . . . . . . 8
8 raaanv 3738 . . . . . . . 8
96, 7, 83bitr4i 270 . . . . . . 7
10 bibi1 319 . . . . . . . . . . . . 13
1110biimpa 472 . . . . . . . . . . . 12
1211biimpcd 217 . . . . . . . . . . 11
1312ralimdv 2787 . . . . . . . . . 10
1413com12 30 . . . . . . . . 9
15 dfss3 3340 . . . . . . . . 9
1614, 15syl6ibr 220 . . . . . . . 8
1716ralimi 2783 . . . . . . 7
189, 17sylbi 189 . . . . . 6
19 eleq1 2498 . . . . . . . . . . 11
2019imbi1d 310 . . . . . . . . . 10
2120rspcv 3050 . . . . . . . . 9
22 dfss3 3340 . . . . . . . . . . 11
2322imbi2i 305 . . . . . . . . . 10
24 r19.21v 2795 . . . . . . . . . 10
2523, 24bitr4i 245 . . . . . . . . 9
2621, 25syl6ib 219 . . . . . . . 8
27 ssel 3344 . . . . . . . . . . 11
2827com12 30 . . . . . . . . . 10
2928imim2d 51 . . . . . . . . 9
3029ralimdv 2787 . . . . . . . 8
3126, 30jcad 521 . . . . . . 7
32 ralbiim 2845 . . . . . . 7
3331, 32syl6ibr 220 . . . . . 6
3418, 33impbid2 197 . . . . 5
3534pm5.32i 620 . . . 4
3635rexbii 2732 . . 3
3736ralbii 2731 . 2
382, 37syl6bb 254 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707  wrex 2708   cin 3321   wss 3322  cuni 4017  cfv 5456  ctop 16960  ccld 17082 This theorem is referenced by:  conpcon  24924 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-topgen 13669  df-top 16965  df-cld 17085
 Copyright terms: Public domain W3C validator