Theorem ishaus 7780

Description: Express the predicate "J is a Hausdorff space."

Hypothesis

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

Assertion

Ref	Expression
ishaus	|- (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) = (/)))))

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

Proof of Theorem ishaus

Step	Hyp	Ref	Expression
1		unieq 2514	. . . 4 |- (j = J -> U.j = U.J)
2		ishaus.1	. . . 4 |- X = U.J
3	1, 2	syl6eqr 1528	. . 3 |- (j = J -> U.j = X)
4		rexeq1 1790	. . . . . 6 |- (j = J -> (E.m e. j (x e. n /\ y e. m /\ (n i^i m) = (/)) <-> E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/))))
5	4	rexeqd 1795	. . . . 5 |- (j = J -> (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 (x e. n /\ y e. m /\ (n i^i m) = (/))))
6	5	imbi2d 614	. . . 4 |- (j = J -> ((x =/= y -> E.n e. j E.m e. j (x e. n /\ y e. m /\ (n i^i m) = (/))) <-> (x =/= y -> E.n e. J E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/)))))
7	3, 6	raleq12d 1797	. . 3 |- (j = J -> (A.y e. U.j(x =/= y -> E.n e. j E.m e. j (x e. n /\ y e. m /\ (n i^i m) = (/))) <-> A.y e. X (x =/= y -> E.n e. J E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/)))))
8	3, 7	raleq12d 1797	. 2 |- (j = J -> (A.x e. U.jA.y e. U.j(x =/= y -> E.n e. j E.m e. j (x e. n /\ y e. m /\ (n i^i m) = (/))) <-> 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) = (/)))))
9		df-haus 7779	. 2 |- Haus = {j e. Top | A.x e. U.jA.y e. U.j(x =/= y -> E.n e. j E.m e. j (x e. n /\ y e. m /\ (n i^i m) = (/)))}
10	8, 9	elrab2 1910	1 |- (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) = (/)))))

Syntax hints:   -> wi 3   <-> wb 146   /\ 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:  hausnei 7781  ishausi 7782  haustop 7783  dtt2 10589

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-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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