Theorem distop 14590

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 3871	. . . . . 6 |- ( x C_ ~P A -> U. x C_ U. ~P A )
2		unipw 4262	. . . . . 6 |- U. ~P A = A
3	1, 2	sseqtrdi 3241	. . . . 5 |- ( x C_ ~P A -> U. x C_ A )
4		vuniex 4486	. . . . . 6 |- U. x e. _V
5	4	elpw 3622	. . . . 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 1472	. . 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 3623	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
10		velpw 3623	. . . . . . . 8 |- ( y e. ~P A <-> y C_ A )
11		ssinss1 3402	. . . . . . . . . 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 2775	. . . . . . . . . . 11 |- y e. _V
14	13	inex2 4180	. . . . . . . . . 10 |- ( x i^i y ) e. _V
15	14	elpw 3622	. . . . . . . . 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 2578	. . . 4 |- ( x e. ~P A -> A. y e. ~P A ( x i^i y ) e. ~P A )
21	20	rgen 2559	. . 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 4225	. . 3 |- ( A e. V -> ~P A e. _V )
24		istopg 14504	. . 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 947	1 |- ( A e. V -> ~P A e. Top )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   e. wcel 2176   A.wral 2484   _Vcvv 2772   i^i cin 3165   C_ wss 3166   ~Pcpw 3616   U.cuni 3850   Topctop 14502

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-un 4481

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-uni 3851  df-top 14503

This theorem is referenced by:  topnex  14591  distopon  14592  distps  14596  discld  14641  restdis  14689  txdis  14782