Theorem distop 19256

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 4259	. . . . . 6 |- ( x C_ ~P A -> U. x C_ U. ~P A )
2		unipw 4690	. . . . . 6 |- U. ~P A = A
3	1, 2	syl6sseq 3543	. . . . 5 |- ( x C_ ~P A -> U. x C_ A )
4		vex 3109	. . . . . . 7 |- x e. _V
5	4	uniex 6571	. . . . . 6 |- U. x e. _V
6	5	elpw 4009	. . . . 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 1596	. . 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 4010	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
11		selpw 4010	. . . . . . . 8 |- ( y e. ~P A <-> y C_ A )
12		ssinss1 3719	. . . . . . . . . 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 3109	. . . . . . . . . . 11 |- y e. _V
15	14	inex2 4582	. . . . . . . . . 10 |- ( x i^i y ) e. _V
16	15	elpw 4009	. . . . . . . . 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 2869	. . . 4 |- ( x e. ~P A -> A. y e. ~P A ( x i^i y ) e. ~P A )
22	21	rgen 2817	. . 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 4624	. . 3 |- ( A e. V -> ~P A e. _V )
25		istopg 19164	. . 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 915	1 |- ( A e. V -> ~P A e. Top )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1372   e. wcel 1762   A.wral 2807   _Vcvv 3106   i^i cin 3468   C_ wss 3469   ~Pcpw 4003   U.cuni 4238   Topctop 19154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-pw 4005  df-sn 4021  df-pr 4023  df-uni 4239  df-top 19159

This theorem is referenced by:  distopon  19257  distps  19275  discld  19349  restdis  19438  dishaus  19642  discmp  19657  dis2ndc  19720  dislly  19757  dis1stc  19759  txdis  19861  xkopt  19884  xkofvcn  19913  symgtgp  20328  locfindis  29764