Theorem distop 15112

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 3954	. . . . . 6 |- ( x C_ ~P A -> U. x C_ U. ~P A )
2		unipw 4355	. . . . . 6 |- U. ~P A = A
3	1, 2	sseqtrdi 3296	. . . . 5 |- ( x C_ ~P A -> U. x C_ A )
4		vuniex 4582	. . . . . 6 |- U. x e. _V
5	4	elpw 3694	. . . . 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 1502	. . 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 3695	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
10		velpw 3695	. . . . . . . 8 |- ( y e. ~P A <-> y C_ A )
11		ssinss1 3460	. . . . . . . . . 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 2824	. . . . . . . . . . 11 |- y e. _V
14	13	inex2 4266	. . . . . . . . . 10 |- ( x i^i y ) e. _V
15	14	elpw 3694	. . . . . . . . 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 2622	. . . 4 |- ( x e. ~P A -> A. y e. ~P A ( x i^i y ) e. ~P A )
21	20	rgen 2603	. . 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 4315	. . 3 |- ( A e. V -> ~P A e. _V )
24		istopg 15026	. . 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 957	1 |- ( A e. V -> ~P A e. Top )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   e. wcel 2209   A.wral 2528   _Vcvv 2821   i^i cin 3219   C_ wss 3220   ~Pcpw 3688   U.cuni 3933   Topctop 15024

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-uni 3934  df-top 15025

This theorem is referenced by:  topnex  15113  distopon  15114  distps  15118  discld  15163  restdis  15211  txdis  15304