Theorem ishausi 7782

Description: Properties that determine a Hausdorff space.

Hypotheses

Ref	Expression
ishausi.1	|- X = U.J
ishausi.2	|- J e. Top
ishausi.3	|- ((x e. X /\ y e. X /\ x =/= y) -> E.n e. J E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/)))

Assertion

Ref	Expression
ishausi	|- J e. Haus

Distinct variable groups:   m,n,x,y,J   x,X,y

Proof of Theorem ishausi

Step	Hyp	Ref	Expression
1		ishausi.1	. . 3 |- X = U.J
2	1	ishaus 7780	. 2 |- (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) = (/)))))
3		ishausi.2	. 2 |- J e. Top
4		ishausi.3	. . . 4 |- ((x e. X /\ y e. X /\ x =/= y) -> E.n e. J E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/)))
5	4	3expia 837	. . 3 |- ((x e. X /\ y e. X) -> (x =/= y -> E.n e. J E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/))))
6	5	rgen2a 1702	. 2 |- 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) = (/)))
7	2, 3, 6	mpbir2an 732	1 |- J e. Haus

Syntax hints:   -> wi 3   /\ 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:  methausi 7878

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-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-uni 2508  df-haus 7779

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