Theorem distop 12293

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 3765	. . . . . 6 |- ( x C_ ~P A -> U. x C_ U. ~P A )
2		unipw 4147	. . . . . 6 |- U. ~P A = A
3	1, 2	sseqtrdi 3150	. . . . 5 |- ( x C_ ~P A -> U. x C_ A )
4		vuniex 4368	. . . . . 6 |- U. x e. _V
5	4	elpw 3521	. . . . 5 |- ( U. x e. ~P A <-> U. x C_ A )
6	3, 5	sylibr 133	. . . 4 |- ( x C_ ~P A -> U. x e. ~P A )
7	6	ax-gen 1426	. . 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 3522	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
10		velpw 3522	. . . . . . . 8 |- ( y e. ~P A <-> y C_ A )
11		ssinss1 3310	. . . . . . . . . 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 2692	. . . . . . . . . . 11 |- y e. _V
14	13	inex2 4071	. . . . . . . . . 10 |- ( x i^i y ) e. _V
15	14	elpw 3521	. . . . . . . . 9 |- ( ( x i^i y ) e. ~P A <-> ( x i^i y ) C_ A )
16	12, 15	syl6ibr 161	. . . . . . . 8 |- ( y C_ A -> ( x C_ A -> ( x i^i y ) e. ~P A ) )
17	10, 16	sylbi 120	. . . . . . 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 120	. . . . 5 |- ( x e. ~P A -> ( y e. ~P A -> ( x i^i y ) e. ~P A ) )
20	19	ralrimiv 2507	. . . 4 |- ( x e. ~P A -> A. y e. ~P A ( x i^i y ) e. ~P A )
21	20	rgen 2488	. . 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 4112	. . 3 |- ( A e. V -> ~P A e. _V )
24		istopg 12205	. . 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 929	1 |- ( A e. V -> ~P A e. Top )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1330   e. wcel 1481   A.wral 2417   _Vcvv 2689   i^i cin 3075   C_ wss 3076   ~Pcpw 3515   U.cuni 3744   Topctop 12203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-uni 3745  df-top 12204

This theorem is referenced by:  topnex  12294  distopon  12295  distps  12299  discld  12344  restdis  12392  txdis  12485