Theorem dtopcl 14948

Description: The open sets of a discrete topology are closed and its closed sets are open.

Hypothesis

Ref	Expression
dtopcl.1	|- A e. _V

Assertion

Ref	Expression
dtopcl	|- ~PA = (Clsd` ~PA)

Proof of Theorem dtopcl

Step	Hyp	Ref	Expression
1		dtopcl.1	. . . 4 |- A e. _V
2	1	distop 8919	. . 3 |- ~PA e. Top
3		eqid 1884	. . . . 5 |- U.~PA = U.~PA
4	3	iscld 8945	. . . 4 |- (~PA e. Top -> (x e. (Clsd` ~PA) <-> (x C_ U.~PA /\ (U.~PA \ x) e. ~PA)))
5		unipw 3504	. . . . . . . . 9 |- U.~PA = A
6	5	sseq2i 2642	. . . . . . . 8 |- (x C_ U.~PA <-> x C_ A)
7	6	biimpi 168	. . . . . . 7 |- (x C_ U.~PA -> x C_ A)
8	1	elpw2 3464	. . . . . . 7 |- (x e. ~PA <-> x C_ A)
9	7, 8	sylibr 217	. . . . . 6 |- (x C_ U.~PA -> x e. ~PA)
10	9	adantr 425	. . . . 5 |- ((x C_ U.~PA /\ (U.~PA \ x) e. ~PA) -> x e. ~PA)
11		elssuni 3206	. . . . . 6 |- (x e. ~PA -> x C_ U.~PA)
12		difss 2735	. . . . . . . 8 |- (U.~PA \ x) C_ U.~PA
13	12, 5	sseqtri 2649	. . . . . . 7 |- (U.~PA \ x) C_ A
14	1	elpw2 3464	. . . . . . 7 |- ((U.~PA \ x) e. ~PA <-> (U.~PA \ x) C_ A)
15	13, 14	mpbir 207	. . . . . 6 |- (U.~PA \ x) e. ~PA
16	11, 15	jctir 317	. . . . 5 |- (x e. ~PA -> (x C_ U.~PA /\ (U.~PA \ x) e. ~PA))
17	10, 16	impbii 174	. . . 4 |- ((x C_ U.~PA /\ (U.~PA \ x) e. ~PA) <-> x e. ~PA)
18	4, 17	syl6rbb 596	. . 3 |- (~PA e. Top -> (x e. ~PA <-> x e. (Clsd` ~PA)))
19	2, 18	ax-mp 7	. 2 |- (x e. ~PA <-> x e. (Clsd` ~PA))
20	19	eqriv 1881	1 |- ~PA = (Clsd` ~PA)

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   \ cdif 2590   C_ wss 2593  ~Pcpw 3032  U.cuni 3177  ` cfv 3998  Topctop 8857  Clsdccld 8936

This theorem is referenced by:  clsingemp 14961

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-top 8861  df-cld 8939

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