Theorem distop 13624

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 3832	. . . . . 6 |- ( x C_ ~P A -> U. x C_ U. ~P A )
2		unipw 4219	. . . . . 6 |- U. ~P A = A
3	1, 2	sseqtrdi 3205	. . . . 5 |- ( x C_ ~P A -> U. x C_ A )
4		vuniex 4440	. . . . . 6 |- U. x e. _V
5	4	elpw 3583	. . . . 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 1449	. . 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 3584	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
10		velpw 3584	. . . . . . . 8 |- ( y e. ~P A <-> y C_ A )
11		ssinss1 3366	. . . . . . . . . 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 2742	. . . . . . . . . . 11 |- y e. _V
14	13	inex2 4140	. . . . . . . . . 10 |- ( x i^i y ) e. _V
15	14	elpw 3583	. . . . . . . . 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 2549	. . . 4 |- ( x e. ~P A -> A. y e. ~P A ( x i^i y ) e. ~P A )
21	20	rgen 2530	. . 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 4182	. . 3 |- ( A e. V -> ~P A e. _V )
24		istopg 13538	. . 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 944	1 |- ( A e. V -> ~P A e. Top )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   e. wcel 2148   A.wral 2455   _Vcvv 2739   i^i cin 3130   C_ wss 3131   ~Pcpw 3577   U.cuni 3811   Topctop 13536

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-uni 3812  df-top 13537

This theorem is referenced by:  topnex  13625  distopon  13626  distps  13630  discld  13675  restdis  13723  txdis  13816