Theorem dtt2 14951

Description: A discrete topology is Hausdorff. Morris, Topology without tears, p.72, ex. 13.

Hypothesis

Ref	Expression
dtt2.1	|- A e. _V

Assertion

Ref	Expression
dtt2	|- ~PA e. Haus

Proof of Theorem dtt2

Step	Hyp	Ref	Expression
1		eqid 1884	. . 3 |- U.~PA = U.~PA
2	1	ishaus 9060	. 2 |- (~PA e. Haus <-> (~PA e. Top /\ A.x e. U.~PAA.y e. U.~PA(x =/= y -> E.u e. ~P AE.v e. ~P A(x e. u /\ y e. v /\ (u i^i v) = (/)))))
3		dtt2.1	. . 3 |- A e. _V
4	3	distop 8919	. 2 |- ~PA e. Top
5		visset 2295	. . . . . . . 8 |- x e. _V
6	5	snid 3069	. . . . . . 7 |- x e. {x}
7	6	a1i 8	. . . . . 6 |- (x =/= y -> x e. {x})
8		visset 2295	. . . . . . . 8 |- y e. _V
9	8	snid 3069	. . . . . . 7 |- y e. {y}
10	9	a1i 8	. . . . . 6 |- (x =/= y -> y e. {y})
11		disjsn2 3091	. . . . . 6 |- (x =/= y -> ({x} i^i {y}) = (/))
12	7, 10, 11	3jca 1050	. . . . 5 |- (x =/= y -> (x e. {x} /\ y e. {y} /\ ({x} i^i {y}) = (/)))
13		eleq2 1958	. . . . . . . . . 10 |- (u = {x} -> (x e. u <-> x e. {x}))
14		ineq1 2789	. . . . . . . . . . 11 |- (u = {x} -> (u i^i v) = ({x} i^i v))
15	14	eqeq1d 1892	. . . . . . . . . 10 |- (u = {x} -> ((u i^i v) = (/) <-> ({x} i^i v) = (/)))
16	13, 15	3anbi13d 1170	. . . . . . . . 9 |- (u = {x} -> ((x e. u /\ y e. v /\ (u i^i v) = (/)) <-> (x e. {x} /\ y e. v /\ ({x} i^i v) = (/))))
17	16	imbi2d 674	. . . . . . . 8 |- (u = {x} -> ((x =/= y -> (x e. u /\ y e. v /\ (u i^i v) = (/))) <-> (x =/= y -> (x e. {x} /\ y e. v /\ ({x} i^i v) = (/)))))
18		eleq2 1958	. . . . . . . . . 10 |- (v = {y} -> (y e. v <-> y e. {y}))
19		ineq2 2790	. . . . . . . . . . 11 |- (v = {y} -> ({x} i^i v) = ({x} i^i {y}))
20	19	eqeq1d 1892	. . . . . . . . . 10 |- (v = {y} -> (({x} i^i v) = (/) <-> ({x} i^i {y}) = (/)))
21	18, 20	3anbi23d 1171	. . . . . . . . 9 |- (v = {y} -> ((x e. {x} /\ y e. v /\ ({x} i^i v) = (/)) <-> (x e. {x} /\ y e. {y} /\ ({x} i^i {y}) = (/))))
22	21	imbi2d 674	. . . . . . . 8 |- (v = {y} -> ((x =/= y -> (x e. {x} /\ y e. v /\ ({x} i^i v) = (/))) <-> (x =/= y -> (x e. {x} /\ y e. {y} /\ ({x} i^i {y}) = (/)))))
23	17, 22	rcla42ev 2385	. . . . . . 7 |- (({x} e. ~PA /\ {y} e. ~PA /\ (x =/= y -> (x e. {x} /\ y e. {y} /\ ({x} i^i {y}) = (/)))) -> E.u e. ~P AE.v e. ~P A(x =/= y -> (x e. u /\ y e. v /\ (u i^i v) = (/))))
24	23	3expia 1069	. . . . . 6 |- (({x} e. ~PA /\ {y} e. ~PA) -> ((x =/= y -> (x e. {x} /\ y e. {y} /\ ({x} i^i {y}) = (/))) -> E.u e. ~P AE.v e. ~P A(x =/= y -> (x e. u /\ y e. v /\ (u i^i v) = (/)))))
25		unipw 3504	. . . . . . . . 9 |- U.~PA = A
26	25	eleq2i 1961	. . . . . . . 8 |- (x e. U.~PA <-> x e. A)
27	26	biimpi 168	. . . . . . 7 |- (x e. U.~PA -> x e. A)
28	5	snelpw 3501	. . . . . . 7 |- (x e. A <-> {x} e. ~PA)
29	27, 28	sylib 215	. . . . . 6 |- (x e. U.~PA -> {x} e. ~PA)
30	25	eleq2i 1961	. . . . . . . 8 |- (y e. U.~PA <-> y e. A)
31	30	biimpi 168	. . . . . . 7 |- (y e. U.~PA -> y e. A)
32	8	snelpw 3501	. . . . . . 7 |- (y e. A <-> {y} e. ~PA)
33	31, 32	sylib 215	. . . . . 6 |- (y e. U.~PA -> {y} e. ~PA)
34	24, 29, 33	syl2an 503	. . . . 5 |- ((x e. U.~PA /\ y e. U.~PA) -> ((x =/= y -> (x e. {x} /\ y e. {y} /\ ({x} i^i {y}) = (/))) -> E.u e. ~P AE.v e. ~P A(x =/= y -> (x e. u /\ y e. v /\ (u i^i v) = (/)))))
35	12, 34	mpi 55	. . . 4 |- ((x e. U.~PA /\ y e. U.~PA) -> E.u e. ~P AE.v e. ~P A(x =/= y -> (x e. u /\ y e. v /\ (u i^i v) = (/))))
36		r19.37av 2233	. . . . 5 |- (E.v e. ~P A(x =/= y -> (x e. u /\ y e. v /\ (u i^i v) = (/))) -> (x =/= y -> E.v e. ~P A(x e. u /\ y e. v /\ (u i^i v) = (/))))
37	36	reximi 2198	. . . 4 |- (E.u e. ~P AE.v e. ~P A(x =/= y -> (x e. u /\ y e. v /\ (u i^i v) = (/))) -> E.u e. ~P A(x =/= y -> E.v e. ~P A(x e. u /\ y e. v /\ (u i^i v) = (/))))
38		r19.37av 2233	. . . 4 |- (E.u e. ~P A(x =/= y -> E.v e. ~P A(x e. u /\ y e. v /\ (u i^i v) = (/))) -> (x =/= y -> E.u e. ~P AE.v e. ~P A(x e. u /\ y e. v /\ (u i^i v) = (/))))
39	35, 37, 38	3syl 24	. . 3 |- ((x e. U.~PA /\ y e. U.~PA) -> (x =/= y -> E.u e. ~P AE.v e. ~P A(x e. u /\ y e. v /\ (u i^i v) = (/))))
40	39	rgen2a 2160	. 2 |- A.x e. U.~PAA.y e. U.~PA(x =/= y -> E.u e. ~P AE.v e. ~P A(x e. u /\ y e. v /\ (u i^i v) = (/)))
41	2, 4, 40	mpbir2an 800	1 |- ~PA e. Haus

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   i^i cin 2592  (/)c0 2875  ~Pcpw 3032  {csn 3044  U.cuni 3177  Topctop 8857  Hauscha 9058

This theorem is referenced by:  dtt1 14952

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-uni 3178  df-top 8861  df-haus 9059

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