Theorem distop 7591

Description: The discrete topology on a set A. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 18-Jul-2006.)

Hypothesis

Ref	Expression
indistop.1	|- A e. V

Assertion

Ref	Expression
distop	|- P~A e. Top

Proof of Theorem distop

Step	Hyp	Ref	Expression
1		indistop.1	. . . 4 |- A e. V
2	1	pwex 2735	. . 3 |- P~A e. V
3		istopg 7538	. . 3 |- (P~A e. V -> (P~A e. Top <-> (A.x(x (_ P~A -> U.x e. P~A) /\ A.x e. P~ AA.y e. P~ A(x i^i y) e. P~A)))
4	2, 3	ax-mp 7	. 2 |- (P~A e. Top <-> (A.x(x (_ P~A -> U.x e. P~A) /\ A.x e. P~ AA.y e. P~ A(x i^i y) e. P~A))
5		uniss 2511	. . . . 5 |- (x (_ P~A -> U.x (_ U.P~A)
6		unipw 2746	. . . . 5 |- U.P~A = A
7	5, 6	syl6ss 2097	. . . 4 |- (x (_ P~A -> U.x (_ A)
8		visset 1804	. . . . . 6 |- x e. V
9	8	uniex 2861	. . . . 5 |- U.x e. V
10	9	elpw 2394	. . . 4 |- (U.x e. P~A <-> U.x (_ A)
11	7, 10	sylibr 200	. . 3 |- (x (_ P~A -> U.x e. P~A)
12	11	ax-gen 960	. 2 |- A.x(x (_ P~A -> U.x e. P~A)
13	8	elpw 2394	. . . . 5 |- (x e. P~A <-> x (_ A)
14		visset 1804	. . . . . . . 8 |- y e. V
15	14	elpw 2394	. . . . . . 7 |- (y e. P~A <-> y (_ A)
16		ssinss1 2227	. . . . . . . . 9 |- (x (_ A -> (x i^i y) (_ A)
17	16	a1i 8	. . . . . . . 8 |- (y (_ A -> (x (_ A -> (x i^i y) (_ A))
18	14	inex2 2707	. . . . . . . . 9 |- (x i^i y) e. V
19	18	elpw 2394	. . . . . . . 8 |- ((x i^i y) e. P~A <-> (x i^i y) (_ A)
20	17, 19	syl6ibr 213	. . . . . . 7 |- (y (_ A -> (x (_ A -> (x i^i y) e. P~A))
21	15, 20	sylbi 199	. . . . . 6 |- (y e. P~A -> (x (_ A -> (x i^i y) e. P~A))
22	21	com12 11	. . . . 5 |- (x (_ A -> (y e. P~A -> (x i^i y) e. P~A))
23	13, 22	sylbi 199	. . . 4 |- (x e. P~A -> (y e. P~A -> (x i^i y) e. P~A))
24	23	r19.21aiv 1705	. . 3 |- (x e. P~A -> A.y e. P~ A(x i^i y) e. P~A)
25	24	rgen 1690	. 2 |- A.x e. P~ AA.y e. P~ A(x i^i y) e. P~A
26	4, 12, 25	mpbir2an 728	1 |- P~A e. Top

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   e. wcel 955  A.wral 1637  Vcvv 1802   i^i cin 2036   (_ wss 2037  P~cpw 2391  U.cuni 2493  Topctop 7530

This theorem is referenced by:  distps 7596  mapdiscn 10398  dtopcl 10459  dtt2 10462

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-uni 2494  df-top 7534

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