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Theorem clsndisj 17144
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 958 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  J  e.  Top )
2 simp2 959 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  S  C_  X
)
3 clscld.1 . . . . . 6  |-  X  = 
U. J
43clsss3 17128 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
54sseld 3349 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  ->  P  e.  X ) )
653impia 1151 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  P  e.  X )
7 simp3 960 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  P  e.  ( ( cls `  J
) `  S )
)
83elcls 17142 . . . 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 1188 . 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 3537 . . . . . 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 313 . . . 4  |-  ( x  =  U  ->  (
( P  e.  x  ->  ( x  i^i  S
)  =/=  (/) )  <->  ( P  e.  U  ->  ( U  i^i  S )  =/=  (/) ) ) )
1514rspccv 3051 . . 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 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    i^i cin 3321    C_ wss 3322   (/)c0 3630   U.cuni 4017   ` cfv 5457   Topctop 16963   clsccl 17087
This theorem is referenced by:  neindisj  17186  clscon  17498  txcls  17641  ptclsg  17652  flimsncls  18023  hauspwpwf1  18024  met2ndci  18557  metdseq0  18889  heibor1lem  26532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-top 16968  df-cld 17088  df-ntr 17089  df-cls 17090
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