Theorem clsndisj 20879

Description: Any open set containing a point that belongs to the closure of a subset intersects the subset. One direction of Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.)

Hypothesis

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

Assertion

Ref	Expression
clsndisj	|- ( ( ( J e. Top /\ S C_ X /\ P e. ( ( cls ` J ) ` S ) ) /\ ( U e. J /\ P e. U ) ) -> ( U i^i S ) =/= (/) )

Proof of Theorem clsndisj

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( J e. Top /\ S C_ X /\ P e. ( ( cls ` J ) ` S ) ) -> J e. Top )
2		simp2 1062	. . 3 |- ( ( J e. Top /\ S C_ X /\ P e. ( ( cls ` J ) ` S ) ) -> S C_ X )
3		clscld.1	. . . . . 6 |- X = U. J
4	3	clsss3 20863	. . . . 5 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ X )
5	4	sseld 3602	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( P e. ( ( cls ` J ) ` S ) -> P e. X ) )
6	5	3impia 1261	. . 3 |- ( ( J e. Top /\ S C_ X /\ P e. ( ( cls ` J ) ` S ) ) -> P e. X )
7		simp3 1063	. . 3 |- ( ( J e. Top /\ S C_ X /\ P e. ( ( cls ` J ) ` S ) ) -> P e. ( ( cls ` J ) ` S ) )
8	3	elcls 20877	. . . 4 |- ( ( J e. Top /\ S C_ X /\ P e. X ) -> ( P e. ( ( cls ` J ) ` S ) <-> A. x e. J ( P e. x -> ( x i^i S ) =/= (/) ) ) )
9	8	biimpa 501	. . 3 |- ( ( ( J e. Top /\ S C_ X /\ P e. X ) /\ P e. ( ( cls ` J ) ` S ) ) -> A. x e. J ( P e. x -> ( x i^i S ) =/= (/) ) )
10	1, 2, 6, 7, 9	syl31anc 1329	. 2 |- ( ( J e. Top /\ S C_ X /\ P e. ( ( cls ` J ) ` S ) ) -> A. x e. J ( P e. x -> ( x i^i S ) =/= (/) ) )
11		eleq2 2690	. . . . 5 |- ( x = U -> ( P e. x <-> P e. U ) )
12		ineq1 3807	. . . . . 6 |- ( x = U -> ( x i^i S ) = ( U i^i S ) )
13	12	neeq1d 2853	. . . . 5 |- ( x = U -> ( ( x i^i S ) =/= (/) <-> ( U i^i S ) =/= (/) ) )
14	11, 13	imbi12d 334	. . . 4 |- ( x = U -> ( ( P e. x -> ( x i^i S ) =/= (/) ) <-> ( P e. U -> ( U i^i S ) =/= (/) ) ) )
15	14	rspccv 3306	. . 3 |- ( A. x e. J ( P e. x -> ( x i^i S ) =/= (/) ) -> ( U e. J -> ( P e. U -> ( U i^i S ) =/= (/) ) ) )
16	15	imp32 449	. 2 |- ( ( A. x e. J ( P e. x -> ( x i^i S ) =/= (/) ) /\ ( U e. J /\ P e. U ) ) -> ( U i^i S ) =/= (/) )
17	10, 16	sylan 488	1 |- ( ( ( J e. Top /\ S C_ X /\ P e. ( ( cls ` J ) ` S ) ) /\ ( U e. J /\ P e. U ) ) -> ( U i^i S ) =/= (/) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436   ` cfv 5888   Topctop 20698   clsccl 20822

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-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-top 20699  df-cld 20823  df-ntr 20824  df-cls 20825

This theorem is referenced by:  neindisj  20921  clsconn  21233  txcls  21407  ptclsg  21418  flimsncls  21790  hauspwpwf1  21791  met2ndci  22327  metdseq0  22657  heibor1lem  33608