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Theorem discld 19463
 Description: The open sets of a discrete topology are closed and its closed sets are open. (Contributed by FL, 7-Jun-2007.) (Revised by Mario Carneiro, 7-Apr-2015.)
Assertion
Ref Expression
discld

Proof of Theorem discld
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 distop 19370 . . . . 5
2 unipw 4687 . . . . . . 7
32eqcomi 2456 . . . . . 6
43iscld 19401 . . . . 5
51, 4syl 16 . . . 4
6 difss 3616 . . . . . 6
7 elpw2g 4600 . . . . . 6
86, 7mpbiri 233 . . . . 5
98biantrud 507 . . . 4
105, 9bitr4d 256 . . 3
11 selpw 4004 . . 3
1210, 11syl6bbr 263 . 2
1312eqrdv 2440 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1383   wcel 1804   cdif 3458   wss 3461  cpw 3997  cuni 4234  cfv 5578  ctop 19267  ccld 19390 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-top 19272  df-cld 19393 This theorem is referenced by:  sn0cld  19464
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