Theorem supnfcls 21824

Description: The filter of supersets of X \ U does not cluster at any point of the open set U. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
supnfcls	|- ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) -> -. A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) )

Distinct variable groups:   x, J   x, X   x, U

Allowed substitution hint:   A( x)

Proof of Theorem supnfcls

Step	Hyp	Ref	Expression
1		disjdif 4040	. 2 |- ( U i^i ( X \ U ) ) = (/)
2		simpr 477	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) /\ A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) ) -> A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) )
3		simpl2 1065	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) /\ A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) ) -> U e. J )
4		simpl3 1066	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) /\ A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) ) -> A e. U )
5		difss 3737	. . . . . . 7 |- ( X \ U ) C_ X
6		simpl1 1064	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) /\ A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) ) -> J e. (TopOn ` X ) )
7		toponmax 20730	. . . . . . . 8 |- ( J e. (TopOn ` X ) -> X e. J )
8		elpw2g 4827	. . . . . . . 8 |- ( X e. J -> ( ( X \ U ) e. ~P X <-> ( X \ U ) C_ X ) )
9	6, 7, 8	3syl 18	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) /\ A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) ) -> ( ( X \ U ) e. ~P X <-> ( X \ U ) C_ X ) )
10	5, 9	mpbiri 248	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) /\ A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) ) -> ( X \ U ) e. ~P X )
11		ssid 3624	. . . . . . 7 |- ( X \ U ) C_ ( X \ U )
12	11	a1i 11	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) /\ A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) ) -> ( X \ U ) C_ ( X \ U ) )
13		sseq2 3627	. . . . . . 7 |- ( x = ( X \ U ) -> ( ( X \ U ) C_ x <-> ( X \ U ) C_ ( X \ U ) ) )
14	13	elrab 3363	. . . . . 6 |- ( ( X \ U ) e. { x e. ~P X | ( X \ U ) C_ x } <-> ( ( X \ U ) e. ~P X /\ ( X \ U ) C_ ( X \ U ) ) )
15	10, 12, 14	sylanbrc 698	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) /\ A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) ) -> ( X \ U ) e. { x e. ~P X | ( X \ U ) C_ x } )
16		fclsopni 21819	. . . . 5 |- ( ( A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) /\ ( U e. J /\ A e. U /\ ( X \ U ) e. { x e. ~P X | ( X \ U ) C_ x } ) ) -> ( U i^i ( X \ U ) ) =/= (/) )
17	2, 3, 4, 15, 16	syl13anc 1328	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) /\ A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) ) -> ( U i^i ( X \ U ) ) =/= (/) )
18	17	ex 450	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) -> ( A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) -> ( U i^i ( X \ U ) ) =/= (/) ) )
19	18	necon2bd 2810	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) -> ( ( U i^i ( X \ U ) ) = (/) -> -. A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) ) )
20	1, 19	mpi 20	1 |- ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. U ) -> -. A e. ( J fClus { x e. ~P X | ( X \ U ) C_ x } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   ` cfv 5888  (class class class)co 6650  TopOnctopon 20715   fClus cfcls 21740

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-rep 4771  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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-fil 21650  df-fcls 21745

This theorem is referenced by:  fclscf  21829