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Theorem clsndisj 18659
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.
StepHypRef Expression
1 simp1 988 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  J  e.  Top )
2 simp2 989 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  S  C_  X
)
3 clscld.1 . . . . . 6  |-  X  = 
U. J
43clsss3 18643 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
54sseld 3350 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  ->  P  e.  X ) )
653impia 1184 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  P  e.  X )
7 simp3 990 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  P  e.  ( ( cls `  J
) `  S )
)
83elcls 18657 . . . 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 )  =/=  (/) ) ) )
98biimpa 484 . . 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 )  =/=  (/) ) )
101, 2, 6, 7, 9syl31anc 1221 . 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 2499 . . . . 5  |-  ( x  =  U  ->  ( P  e.  x  <->  P  e.  U ) )
12 ineq1 3540 . . . . . 6  |-  ( x  =  U  ->  (
x  i^i  S )  =  ( U  i^i  S ) )
1312neeq1d 2616 . . . . 5  |-  ( x  =  U  ->  (
( x  i^i  S
)  =/=  (/)  <->  ( U  i^i  S )  =/=  (/) ) )
1411, 13imbi12d 320 . . . 4  |-  ( x  =  U  ->  (
( P  e.  x  ->  ( x  i^i  S
)  =/=  (/) )  <->  ( P  e.  U  ->  ( U  i^i  S )  =/=  (/) ) ) )
1514rspccv 3065 . . 3  |-  ( A. x  e.  J  ( P  e.  x  ->  ( x  i^i  S )  =/=  (/) )  ->  ( U  e.  J  ->  ( P  e.  U  -> 
( U  i^i  S
)  =/=  (/) ) ) )
1615imp32 433 . 2  |-  ( ( A. x  e.  J  ( P  e.  x  ->  ( x  i^i  S
)  =/=  (/) )  /\  ( U  e.  J  /\  P  e.  U
) )  ->  ( U  i^i  S )  =/=  (/) )
1710, 16sylan 471 1  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) )  /\  ( U  e.  J  /\  P  e.  U
) )  ->  ( U  i^i  S )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710    i^i cin 3322    C_ wss 3323   (/)c0 3632   U.cuni 4086   ` cfv 5413   Topctop 18478   clsccl 18602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-top 18483  df-cld 18603  df-ntr 18604  df-cls 18605
This theorem is referenced by:  neindisj  18701  clscon  19014  txcls  19157  ptclsg  19168  flimsncls  19539  hauspwpwf1  19540  met2ndci  20077  metdseq0  20410  heibor1lem  28682
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