Theorem distop 7542

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 2713	. . 3 |- P~A e. V
3		istopg 7489	. . 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 2489	. . . . 5 |- (x (_ P~A -> U.x (_ U.P~A)
6		unipw 2724	. . . . 5 |- U.P~A = A
7	5, 6	syl6ss 2078	. . . 4 |- (x (_ P~A -> U.x (_ A)
8		visset 1788	. . . . . 6 |- x e. V
9	8	uniex 2834	. . . . 5 |- U.x e. V
10	9	elpw 2375	. . . 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 955	. 2 |- A.x(x (_ P~A -> U.x e. P~A)
13	8	elpw 2375	. . . . 5 |- (x e. P~A <-> x (_ A)
14		visset 1788	. . . . . . . 8 |- y e. V
15	14	elpw 2375	. . . . . . 7 |- (y e. P~A <-> y (_ A)
16		ssinss1 2208	. . . . . . . . 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 2685	. . . . . . . . 9 |- (x i^i y) e. V
19	18	elpw 2375	. . . . . . . 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 1689	. . 3 |- (x e. P~A -> A.y e. P~ A(x i^i y) e. P~A)
25	24	rgen 1674	. 2 |- A.x e. P~ AA.y e. P~ A(x i^i y) e. P~A
26	4, 12, 25	mpbir2an 727	1 |- P~A e. Top

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 950   e. wcel 1105  A.wral 1621  Vcvv 1786   i^i cin 2017   (_ wss 2018  P~cpw 2372  U.cuni 2471  Topctop 7481

This theorem is referenced by:  distps 7547  dtopcl 8809  dtt2 8812

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-pow 2710  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-uni 2472  df-top 7485

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