Theorem hausnei 7763

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 1589	. . . . . . 7 |- (x = P -> (x =/= y <-> P =/= y))
2		eleq1 1533	. . . . . . . . 9 |- (x = P -> (x e. n <-> P e. n))
3	2	3anbi1d 896	. . . . . . . 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 1680	. . . . . . 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 625	. . . . . 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 1590	. . . . . . 7 |- (y = Q -> (P =/= y <-> P =/= Q))
7		eleq1 1533	. . . . . . . . 9 |- (y = Q -> (y e. m <-> Q e. m))
8	7	3anbi2d 897	. . . . . . . 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 1680	. . . . . . 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 625	. . . . . 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 1878	. . . . 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 7762	. . . . . 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 847	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 774   = wceq 955   e. wcel 957   =/= wne 1584  A.wral 1644  E.wrex 1645   i^i cin 2044  (/)c0 2278  U.cuni 2500  Topctop 7567  Hauscha 7760

This theorem is referenced by:  sncld 7766

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-rab 1651  df-v 1810  df-uni 2501  df-haus 7761

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