Theorem hausnei 9061

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 2024	. . . . . . 7 |- (x = P -> (x =/= y <-> P =/= y))
2		eleq1 1957	. . . . . . . . 9 |- (x = P -> (x e. n <-> P e. n))
3	2	3anbi1d 1172	. . . . . . . 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 2141	. . . . . . 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 688	. . . . . 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 2025	. . . . . . 7 |- (y = Q -> (P =/= y <-> P =/= Q))
7		eleq1 1957	. . . . . . . . 9 |- (y = Q -> (y e. m <-> Q e. m))
8	7	3anbi2d 1173	. . . . . . . 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 2141	. . . . . . 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 688	. . . . . 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 2384	. . . . 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 9060	. . . . . 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	simprbi 353	. . . . 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 20	. . . 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 402	. . 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 39	. 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 1083	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 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   i^i cin 2592  (/)c0 2875  U.cuni 3177  Topctop 8857  Hauscha 9058

This theorem is referenced by:  sncld 9064  hscptsscld 15434  haustlmu 15906

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-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

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-uni 3178  df-haus 9059

Copyright terms: Public domain