Theorem distop 8919

Description: The discrete topology on a set A. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 18-Jul-2006.)

Hypothesis

Ref	Expression
indistop.1	|- A e. _V

Assertion

Ref	Expression
distop	|- ~PA e. Top

Proof of Theorem distop

Step	Hyp	Ref	Expression
1		indistop.1	. . . 4 |- A e. _V
2	1	pwex 3487	. . 3 |- ~PA e. _V
3		istopg 8865	. . 3 |- (~PA e. _V -> (~PA e. Top <-> (A.x(x C_ ~PA -> U.x e. ~PA) /\ A.x e. ~P AA.y e. ~P A(x i^i y) e. ~PA)))
4	2, 3	ax-mp 7	. 2 |- (~PA e. Top <-> (A.x(x C_ ~PA -> U.x e. ~PA) /\ A.x e. ~P AA.y e. ~P A(x i^i y) e. ~PA))
5		uniss 3199	. . . . 5 |- (x C_ ~PA -> U.x C_ U.~PA)
6		unipw 3504	. . . . 5 |- U.~PA = A
7	5, 6	syl6ss 2663	. . . 4 |- (x C_ ~PA -> U.x C_ A)
8		visset 2295	. . . . . 6 |- x e. _V
9	8	uniex 3794	. . . . 5 |- U.x e. _V
10	9	elpw 3037	. . . 4 |- (U.x e. ~PA <-> U.x C_ A)
11	7, 10	sylibr 217	. . 3 |- (x C_ ~PA -> U.x e. ~PA)
12	11	ax-gen 1305	. 2 |- A.x(x C_ ~PA -> U.x e. ~PA)
13	8	elpw 3037	. . . . 5 |- (x e. ~PA <-> x C_ A)
14		visset 2295	. . . . . . . 8 |- y e. _V
15	14	elpw 3037	. . . . . . 7 |- (y e. ~PA <-> y C_ A)
16		ssinss1 2821	. . . . . . . . 9 |- (x C_ A -> (x i^i y) C_ A)
17	16	a1i 8	. . . . . . . 8 |- (y C_ A -> (x C_ A -> (x i^i y) C_ A))
18	14	inex2 3453	. . . . . . . . 9 |- (x i^i y) e. _V
19	18	elpw 3037	. . . . . . . 8 |- ((x i^i y) e. ~PA <-> (x i^i y) C_ A)
20	17, 19	syl6ibr 230	. . . . . . 7 |- (y C_ A -> (x C_ A -> (x i^i y) e. ~PA))
21	15, 20	sylbi 216	. . . . . 6 |- (y e. ~PA -> (x C_ A -> (x i^i y) e. ~PA))
22	21	com12 14	. . . . 5 |- (x C_ A -> (y e. ~PA -> (x i^i y) e. ~PA))
23	13, 22	sylbi 216	. . . 4 |- (x e. ~PA -> (y e. ~PA -> (x i^i y) e. ~PA))
24	23	r19.21aiv 2175	. . 3 |- (x e. ~PA -> A.y e. ~P A(x i^i y) e. ~PA)
25	24	rgen 2159	. 2 |- A.x e. ~P AA.y e. ~P A(x i^i y) e. ~PA
26	4, 12, 25	mpbir2an 800	1 |- ~PA e. Top

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300  A.wral 2105  _Vcvv 2292   i^i cin 2592   C_ wss 2593  ~Pcpw 3032  U.cuni 3177  Topctop 8857

This theorem is referenced by:  distps 8924  mapdiscnlem 14870  distopg 14876  dtopcl 14948  dtt2 14951  locfindsc 15515  topmtcl 15525

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-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-uni 3178  df-top 8861

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