Theorem distop 17015

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 3996	. . . . . 6 |- ( x C_ ~P A -> U. x C_ U. ~P A )
2		unipw 4374	. . . . . 6 |- U. ~P A = A
3	1, 2	syl6sseq 3354	. . . . 5 |- ( x C_ ~P A -> U. x C_ A )
4		vex 2919	. . . . . . 7 |- x e. _V
5	4	uniex 4664	. . . . . 6 |- U. x e. _V
6	5	elpw 3765	. . . . 5 |- ( U. x e. ~P A <-> U. x C_ A )
7	3, 6	sylibr 204	. . . 4 |- ( x C_ ~P A -> U. x e. ~P A )
8	7	ax-gen 1552	. . 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	4	elpw 3765	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
11		vex 2919	. . . . . . . . 9 |- y e. _V
12	11	elpw 3765	. . . . . . . 8 |- ( y e. ~P A <-> y C_ A )
13		ssinss1 3529	. . . . . . . . . 10 |- ( x C_ A -> ( x i^i y ) C_ A )
14	13	a1i 11	. . . . . . . . 9 |- ( y C_ A -> ( x C_ A -> ( x i^i y ) C_ A ) )
15	11	inex2 4305	. . . . . . . . . 10 |- ( x i^i y ) e. _V
16	15	elpw 3765	. . . . . . . . 9 |- ( ( x i^i y ) e. ~P A <-> ( x i^i y ) C_ A )
17	14, 16	syl6ibr 219	. . . . . . . 8 |- ( y C_ A -> ( x C_ A -> ( x i^i y ) e. ~P A ) )
18	12, 17	sylbi 188	. . . . . . 7 |- ( y e. ~P A -> ( x C_ A -> ( x i^i y ) e. ~P A ) )
19	18	com12 29	. . . . . 6 |- ( x C_ A -> ( y e. ~P A -> ( x i^i y ) e. ~P A ) )
20	10, 19	sylbi 188	. . . . 5 |- ( x e. ~P A -> ( y e. ~P A -> ( x i^i y ) e. ~P A ) )
21	20	ralrimiv 2748	. . . 4 |- ( x e. ~P A -> A. y e. ~P A ( x i^i y ) e. ~P A )
22	21	rgen 2731	. . 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 4343	. . 3 |- ( A e. V -> ~P A e. _V )
25		istopg 16923	. . 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 889	1 |- ( A e. V -> ~P A e. Top )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   A.wal 1546   e. wcel 1721   A.wral 2666   _Vcvv 2916   i^i cin 3279   C_ wss 3280   ~Pcpw 3759   U.cuni 3975   Topctop 16913

This theorem is referenced by:  distopon  17016  distps  17034  discld  17108  restdis  17196  dishaus  17400  discmp  17415  dis2ndc  17476  dislly  17513  dis1stc  17515  txdis  17617  xkopt  17640  xkofvcn  17669  symgtgp  18084  locfindis  26275

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-pw 3761  df-sn 3780  df-pr 3781  df-uni 3976  df-top 16918