Theorem distop 18600

Description: The discrete topology on a set A. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
distop	|- ( A e. V -> ~P A e. Top )

Proof of Theorem distop

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniss 4112	. . . . . 6 |- ( x C_ ~P A -> U. x C_ U. ~P A )
2		unipw 4542	. . . . . 6 |- U. ~P A = A
3	1, 2	syl6sseq 3402	. . . . 5 |- ( x C_ ~P A -> U. x C_ A )
4		vex 2975	. . . . . . 7 |- x e. _V
5	4	uniex 6376	. . . . . 6 |- U. x e. _V
6	5	elpw 3866	. . . . 5 |- ( U. x e. ~P A <-> U. x C_ A )
7	3, 6	sylibr 212	. . . 4 |- ( x C_ ~P A -> U. x e. ~P A )
8	7	ax-gen 1591	. . 3 |- A. x ( x C_ ~P A -> U. x e. ~P A )
9	8	a1i 11	. 2 |- ( A e. V -> A. x ( x C_ ~P A -> U. x e. ~P A ) )
10		selpw 3867	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
11		selpw 3867	. . . . . . . 8 |- ( y e. ~P A <-> y C_ A )
12		ssinss1 3578	. . . . . . . . . 10 |- ( x C_ A -> ( x i^i y ) C_ A )
13	12	a1i 11	. . . . . . . . 9 |- ( y C_ A -> ( x C_ A -> ( x i^i y ) C_ A ) )
14		vex 2975	. . . . . . . . . . 11 |- y e. _V
15	14	inex2 4434	. . . . . . . . . 10 |- ( x i^i y ) e. _V
16	15	elpw 3866	. . . . . . . . 9 |- ( ( x i^i y ) e. ~P A <-> ( x i^i y ) C_ A )
17	13, 16	syl6ibr 227	. . . . . . . 8 |- ( y C_ A -> ( x C_ A -> ( x i^i y ) e. ~P A ) )
18	11, 17	sylbi 195	. . . . . . 7 |- ( y e. ~P A -> ( x C_ A -> ( x i^i y ) e. ~P A ) )
19	18	com12 31	. . . . . 6 |- ( x C_ A -> ( y e. ~P A -> ( x i^i y ) e. ~P A ) )
20	10, 19	sylbi 195	. . . . 5 |- ( x e. ~P A -> ( y e. ~P A -> ( x i^i y ) e. ~P A ) )
21	20	ralrimiv 2798	. . . 4 |- ( x e. ~P A -> A. y e. ~P A ( x i^i y ) e. ~P A )
22	21	rgen 2781	. . 3 |- A. x e. ~P A A. y e. ~P A ( x i^i y ) e. ~P A
23	22	a1i 11	. 2 |- ( A e. V -> A. x e. ~P A A. y e. ~P A ( x i^i y ) e. ~P A )
24		pwexg 4476	. . 3 |- ( A e. V -> ~P A e. _V )
25		istopg 18508	. . 3 |- ( ~P A e. _V -> ( ~P A e. Top <-> ( A. x ( x C_ ~P A -> U. x e. ~P A ) /\ A. x e. ~P A A. y e. ~P A ( x i^i y ) e. ~P A ) ) )
26	24, 25	syl 16	. 2 |- ( A e. V -> ( ~P A e. Top <-> ( A. x ( x C_ ~P A -> U. x e. ~P A ) /\ A. x e. ~P A A. y e. ~P A ( x i^i y ) e. ~P A ) ) )
27	9, 23, 26	mpbir2and 913	1 |- ( A e. V -> ~P A e. Top )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1367   e. wcel 1756   A.wral 2715   _Vcvv 2972   i^i cin 3327   C_ wss 3328   ~Pcpw 3860   U.cuni 4091   Topctop 18498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-pw 3862  df-sn 3878  df-pr 3880  df-uni 4092  df-top 18503

This theorem is referenced by:  distopon  18601  distps  18619  discld  18693  restdis  18782  dishaus  18986  discmp  19001  dis2ndc  19064  dislly  19101  dis1stc  19103  txdis  19205  xkopt  19228  xkofvcn  19257  symgtgp  19672  locfindis  28577