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Theorem clsndisj 16806
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
StepHypRef Expression
1 simp1 960 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  J  e.  Top )
2 simp2 961 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  S  C_  X
)
3 clscld.1 . . . . . 6  |-  X  = 
U. J
43clsss3 16790 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
54sseld 3180 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  ->  P  e.  X ) )
653impia 1153 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  P  e.  X )
7 simp3 962 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  P  e.  ( ( cls `  J
) `  S )
)
83elcls 16804 . . . 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 472 . . 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 1190 . 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 2345 . . . . 5  |-  ( x  =  U  ->  ( P  e.  x  <->  P  e.  U ) )
12 ineq1 3364 . . . . . 6  |-  ( x  =  U  ->  (
x  i^i  S )  =  ( U  i^i  S ) )
1312neeq1d 2460 . . . . 5  |-  ( x  =  U  ->  (
( x  i^i  S
)  =/=  (/)  <->  ( U  i^i  S )  =/=  (/) ) )
1411, 13imbi12d 313 . . . 4  |-  ( x  =  U  ->  (
( P  e.  x  ->  ( x  i^i  S
)  =/=  (/) )  <->  ( P  e.  U  ->  ( U  i^i  S )  =/=  (/) ) ) )
1514rspccv 2882 . . 3  |-  ( A. x  e.  J  ( P  e.  x  ->  ( x  i^i  S )  =/=  (/) )  ->  ( U  e.  J  ->  ( P  e.  U  -> 
( U  i^i  S
)  =/=  (/) ) ) )
1615imp32 424 . 2  |-  ( ( A. x  e.  J  ( P  e.  x  ->  ( x  i^i  S
)  =/=  (/) )  /\  ( U  e.  J  /\  P  e.  U
) )  ->  ( U  i^i  S )  =/=  (/) )
1710, 16sylan 459 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 set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688    =/= wne 2447   A.wral 2544    i^i cin 3152    C_ wss 3153   (/)c0 3456   U.cuni 3828   ` cfv 5221   Topctop 16625   clsccl 16749
This theorem is referenced by:  neindisj  16848  clscon  17150  txcls  17293  ptclsg  17303  flimsncls  17675  hauspwpwf1  17676  met2ndci  18062  metdseq0  18352  heibor1lem  25932
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-top 16630  df-cld 16750  df-ntr 16751  df-cls 16752
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