Theorem ishausi 7785

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 7783	. 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 835	. . 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 1699	. 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 730	1 |- J e. Haus

Syntax hints:   -> wi 3   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  E.wrex 1646   i^i cin 2046  (/)c0 2280  U.cuni 2503  Topctop 7588  Hauscha 7781

This theorem is referenced by:  methausi 7881

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-uni 2504  df-haus 7782

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