Theorem hausnei 7781

Description: Neighborhood property of a Hausdorff space.

Hypothesis

Ref	Expression
ishaus.1	|- X = U.J

Assertion

Ref	Expression
hausnei	|- ((J e. Haus /\ (P e. X /\ Q e. X /\ P =/= Q)) -> E.n e. J E.m e. J (P e. n /\ Q e. m /\ (n i^i m) = (/)))

Distinct variable groups:   m,n,J   P,m,n   Q,m,n

Proof of Theorem hausnei

Step	Hyp	Ref	Expression
1		neeq1 1593	. . . . . . 7 |- (x = P -> (x =/= y <-> P =/= y))
2		eleq1 1537	. . . . . . . . 9 |- (x = P -> (x e. n <-> P e. n))
3	2	3anbi1d 899	. . . . . . . 8 |- (x = P -> ((x e. n /\ y e. m /\ (n i^i m) = (/)) <-> (P e. n /\ y e. m /\ (n i^i m) = (/))))
4	3	2rexbidv 1684	. . . . . . 7 |- (x = P -> (E.n e. J E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/)) <-> E.n e. J E.m e. J (P e. n /\ y e. m /\ (n i^i m) = (/))))
5	1, 4	imbi12d 628	. . . . . 6 |- (x = P -> ((x =/= y -> E.n e. J E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/))) <-> (P =/= y -> E.n e. J E.m e. J (P e. n /\ y e. m /\ (n i^i m) = (/)))))
6		neeq2 1594	. . . . . . 7 |- (y = Q -> (P =/= y <-> P =/= Q))
7		eleq1 1537	. . . . . . . . 9 |- (y = Q -> (y e. m <-> Q e. m))
8	7	3anbi2d 900	. . . . . . . 8 |- (y = Q -> ((P e. n /\ y e. m /\ (n i^i m) = (/)) <-> (P e. n /\ Q e. m /\ (n i^i m) = (/))))
9	8	2rexbidv 1684	. . . . . . 7 |- (y = Q -> (E.n e. J E.m e. J (P e. n /\ y e. m /\ (n i^i m) = (/)) <-> E.n e. J E.m e. J (P e. n /\ Q e. m /\ (n i^i m) = (/))))
10	6, 9	imbi12d 628	. . . . . 6 |- (y = Q -> ((P =/= y -> E.n e. J E.m e. J (P e. n /\ y e. m /\ (n i^i m) = (/))) <-> (P =/= Q -> E.n e. J E.m e. J (P e. n /\ Q e. m /\ (n i^i m) = (/)))))
11	5, 10	rcla42v 1883	. . . . 5 |- ((P e. X /\ Q e. X) -> (A.x e. X A.y e. X (x =/= y -> E.n e. J E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/))) -> (P =/= Q -> E.n e. J E.m e. J (P e. n /\ Q e. m /\ (n i^i m) = (/)))))
12		ishaus.1	. . . . . . 7 |- X = U.J
13	12	ishaus 7780	. . . . . 6 |- (J e. Haus <-> (J e. Top /\ A.x e. X A.y e. X (x =/= y -> E.n e. J E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/)))))
14	13	pm3.27bi 326	. . . . 5 |- (J e. Haus -> A.x e. X A.y e. X (x =/= y -> E.n e. J E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/))))
15	11, 14	syl5 21	. . . 4 |- ((P e. X /\ Q e. X) -> (J e. Haus -> (P =/= Q -> E.n e. J E.m e. J (P e. n /\ Q e. m /\ (n i^i m) = (/)))))
16	15	ex 373	. . 3 |- (P e. X -> (Q e. X -> (J e. Haus -> (P =/= Q -> E.n e. J E.m e. J (P e. n /\ Q e. m /\ (n i^i m) = (/))))))
17	16	com3r 35	. 2 |- (J e. Haus -> (P e. X -> (Q e. X -> (P =/= Q -> E.n e. J E.m e. J (P e. n /\ Q e. m /\ (n i^i m) = (/))))))
18	17	3imp2 850	1 |- ((J e. Haus /\ (P e. X /\ Q e. X /\ P =/= Q)) -> E.n e. J E.m e. J (P e. n /\ Q e. m /\ (n i^i m) = (/)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648  E.wrex 1649   i^i cin 2049  (/)c0 2283  U.cuni 2507  Topctop 7590  Hauscha 7778

This theorem is referenced by:  sncld 7784

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-uni 2508  df-haus 7779

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