Theorem distop 15076

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 3940	. . . . . 6 |- ( x C_ ~P A -> U. x C_ U. ~P A )
2		unipw 4338	. . . . . 6 |- U. ~P A = A
3	1, 2	sseqtrdi 3290	. . . . 5 |- ( x C_ ~P A -> U. x C_ A )
4		vuniex 4564	. . . . . 6 |- U. x e. _V
5	4	elpw 3680	. . . . 5 |- ( U. x e. ~P A <-> U. x C_ A )
6	3, 5	sylibr 134	. . . 4 |- ( x C_ ~P A -> U. x e. ~P A )
7	6	ax-gen 1498	. . 3 |- A. x ( x C_ ~P A -> U. x e. ~P A )
8	7	a1i 9	. 2 |- ( A e. V -> A. x ( x C_ ~P A -> U. x e. ~P A ) )
9		velpw 3681	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
10		velpw 3681	. . . . . . . 8 |- ( y e. ~P A <-> y C_ A )
11		ssinss1 3454	. . . . . . . . . 10 |- ( x C_ A -> ( x i^i y ) C_ A )
12	11	a1i 9	. . . . . . . . 9 |- ( y C_ A -> ( x C_ A -> ( x i^i y ) C_ A ) )
13		vex 2818	. . . . . . . . . . 11 |- y e. _V
14	13	inex2 4250	. . . . . . . . . 10 |- ( x i^i y ) e. _V
15	14	elpw 3680	. . . . . . . . 9 |- ( ( x i^i y ) e. ~P A <-> ( x i^i y ) C_ A )
16	12, 15	imbitrrdi 162	. . . . . . . 8 |- ( y C_ A -> ( x C_ A -> ( x i^i y ) e. ~P A ) )
17	10, 16	sylbi 121	. . . . . . 7 |- ( y e. ~P A -> ( x C_ A -> ( x i^i y ) e. ~P A ) )
18	17	com12 30	. . . . . 6 |- ( x C_ A -> ( y e. ~P A -> ( x i^i y ) e. ~P A ) )
19	9, 18	sylbi 121	. . . . 5 |- ( x e. ~P A -> ( y e. ~P A -> ( x i^i y ) e. ~P A ) )
20	19	ralrimiv 2616	. . . 4 |- ( x e. ~P A -> A. y e. ~P A ( x i^i y ) e. ~P A )
21	20	rgen 2597	. . 3 |- A. x e. ~P A A. y e. ~P A ( x i^i y ) e. ~P A
22	21	a1i 9	. 2 |- ( A e. V -> A. x e. ~P A A. y e. ~P A ( x i^i y ) e. ~P A )
23		pwexg 4298	. . 3 |- ( A e. V -> ~P A e. _V )
24		istopg 14990	. . 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 ) ) )
25	23, 24	syl 14	. 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 ) ) )
26	8, 22, 25	mpbir2and 953	1 |- ( A e. V -> ~P A e. Top )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1396   e. wcel 2205   A.wral 2522   _Vcvv 2815   i^i cin 3213   C_ wss 3214   ~Pcpw 3674   U.cuni 3919   Topctop 14988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-uni 3920  df-top 14989

This theorem is referenced by:  topnex  15077  distopon  15078  distps  15082  discld  15127  restdis  15175  txdis  15268