Theorem dishaus 21186

Description: A discrete topology is Hausdorff. Morris, Topology without tears, p.72, ex. 13. (Contributed by FL, 24-Jun-2007.) (Proof shortened by Mario Carneiro, 8-Apr-2015.)

Assertion

Ref	Expression
dishaus	|- ( A e. V -> ~P A e. Haus )

Proof of Theorem dishaus

Dummy variables v u x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		distop 20799	. 2 |- ( A e. V -> ~P A e. Top )
2		simplrl 800	. . . . . . 7 |- ( ( ( A e. V /\ ( x e. A /\ y e. A ) ) /\ x =/= y ) -> x e. A )
3	2	snssd 4340	. . . . . 6 |- ( ( ( A e. V /\ ( x e. A /\ y e. A ) ) /\ x =/= y ) -> { x } C_ A )
4		snex 4908	. . . . . . 7 |- { x } e. _V
5	4	elpw 4164	. . . . . 6 |- ( { x } e. ~P A <-> { x } C_ A )
6	3, 5	sylibr 224	. . . . 5 |- ( ( ( A e. V /\ ( x e. A /\ y e. A ) ) /\ x =/= y ) -> { x } e. ~P A )
7		simplrr 801	. . . . . . 7 |- ( ( ( A e. V /\ ( x e. A /\ y e. A ) ) /\ x =/= y ) -> y e. A )
8	7	snssd 4340	. . . . . 6 |- ( ( ( A e. V /\ ( x e. A /\ y e. A ) ) /\ x =/= y ) -> { y } C_ A )
9		snex 4908	. . . . . . 7 |- { y } e. _V
10	9	elpw 4164	. . . . . 6 |- ( { y } e. ~P A <-> { y } C_ A )
11	8, 10	sylibr 224	. . . . 5 |- ( ( ( A e. V /\ ( x e. A /\ y e. A ) ) /\ x =/= y ) -> { y } e. ~P A )
12		vsnid 4209	. . . . . 6 |- x e. { x }
13	12	a1i 11	. . . . 5 |- ( ( ( A e. V /\ ( x e. A /\ y e. A ) ) /\ x =/= y ) -> x e. { x } )
14		vsnid 4209	. . . . . 6 |- y e. { y }
15	14	a1i 11	. . . . 5 |- ( ( ( A e. V /\ ( x e. A /\ y e. A ) ) /\ x =/= y ) -> y e. { y } )
16		disjsn2 4247	. . . . . 6 |- ( x =/= y -> ( { x } i^i { y } ) = (/) )
17	16	adantl 482	. . . . 5 |- ( ( ( A e. V /\ ( x e. A /\ y e. A ) ) /\ x =/= y ) -> ( { x } i^i { y } ) = (/) )
18		eleq2 2690	. . . . . . 7 |- ( u = { x } -> ( x e. u <-> x e. { x } ) )
19		ineq1 3807	. . . . . . . 8 |- ( u = { x } -> ( u i^i v ) = ( { x } i^i v ) )
20	19	eqeq1d 2624	. . . . . . 7 |- ( u = { x } -> ( ( u i^i v ) = (/) <-> ( { x } i^i v ) = (/) ) )
21	18, 20	3anbi13d 1401	. . . . . 6 |- ( u = { x } -> ( ( x e. u /\ y e. v /\ ( u i^i v ) = (/) ) <-> ( x e. { x } /\ y e. v /\ ( { x } i^i v ) = (/) ) ) )
22		eleq2 2690	. . . . . . 7 |- ( v = { y } -> ( y e. v <-> y e. { y } ) )
23		ineq2 3808	. . . . . . . 8 |- ( v = { y } -> ( { x } i^i v ) = ( { x } i^i { y } ) )
24	23	eqeq1d 2624	. . . . . . 7 |- ( v = { y } -> ( ( { x } i^i v ) = (/) <-> ( { x } i^i { y } ) = (/) ) )
25	22, 24	3anbi23d 1402	. . . . . 6 |- ( v = { y } -> ( ( x e. { x } /\ y e. v /\ ( { x } i^i v ) = (/) ) <-> ( x e. { x } /\ y e. { y } /\ ( { x } i^i { y } ) = (/) ) ) )
26	21, 25	rspc2ev 3324	. . . . 5 |- ( ( { x } e. ~P A /\ { y } e. ~P A /\ ( x e. { x } /\ y e. { y } /\ ( { x } i^i { y } ) = (/) ) ) -> E. u e. ~P A E. v e. ~P A ( x e. u /\ y e. v /\ ( u i^i v ) = (/) ) )
27	6, 11, 13, 15, 17, 26	syl113anc 1338	. . . 4 |- ( ( ( A e. V /\ ( x e. A /\ y e. A ) ) /\ x =/= y ) -> E. u e. ~P A E. v e. ~P A ( x e. u /\ y e. v /\ ( u i^i v ) = (/) ) )
28	27	ex 450	. . 3 |- ( ( A e. V /\ ( x e. A /\ y e. A ) ) -> ( x =/= y -> E. u e. ~P A E. v e. ~P A ( x e. u /\ y e. v /\ ( u i^i v ) = (/) ) ) )
29	28	ralrimivva 2971	. 2 |- ( A e. V -> A. x e. A A. y e. A ( x =/= y -> E. u e. ~P A E. v e. ~P A ( x e. u /\ y e. v /\ ( u i^i v ) = (/) ) ) )
30		unipw 4918	. . . 4 |- U. ~P A = A
31	30	eqcomi 2631	. . 3 |- A = U. ~P A
32	31	ishaus 21126	. 2 |- ( ~P A e. Haus <-> ( ~P A e. Top /\ A. x e. A A. y e. A ( x =/= y -> E. u e. ~P A E. v e. ~P A ( x e. u /\ y e. v /\ ( u i^i v ) = (/) ) ) ) )
33	1, 29, 32	sylanbrc 698	1 |- ( A e. V -> ~P A e. Haus )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   Topctop 20698   Hauscha 21112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-top 20699  df-haus 21119

This theorem is referenced by:  ssoninhaus  32447