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Theorem clsndisj 16812
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 955 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  J  e.  Top )
2 simp2 956 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  S  C_  X
)
3 clscld.1 . . . . . 6  |-  X  = 
U. J
43clsss3 16796 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
54sseld 3179 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  ->  P  e.  X ) )
653impia 1148 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  P  e.  X )
7 simp3 957 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  ->  P  e.  ( ( cls `  J
) `  S )
)
83elcls 16810 . . . 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 470 . . 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 1185 . 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 2344 . . . . 5  |-  ( x  =  U  ->  ( P  e.  x  <->  P  e.  U ) )
12 ineq1 3363 . . . . . 6  |-  ( x  =  U  ->  (
x  i^i  S )  =  ( U  i^i  S ) )
1312neeq1d 2459 . . . . 5  |-  ( x  =  U  ->  (
( x  i^i  S
)  =/=  (/)  <->  ( U  i^i  S )  =/=  (/) ) )
1411, 13imbi12d 311 . . . 4  |-  ( x  =  U  ->  (
( P  e.  x  ->  ( x  i^i  S
)  =/=  (/) )  <->  ( P  e.  U  ->  ( U  i^i  S )  =/=  (/) ) ) )
1514rspccv 2881 . . 3  |-  ( A. x  e.  J  ( P  e.  x  ->  ( x  i^i  S )  =/=  (/) )  ->  ( U  e.  J  ->  ( P  e.  U  -> 
( U  i^i  S
)  =/=  (/) ) ) )
1615imp32 422 . 2  |-  ( ( A. x  e.  J  ( P  e.  x  ->  ( x  i^i  S
)  =/=  (/) )  /\  ( U  e.  J  /\  P  e.  U
) )  ->  ( U  i^i  S )  =/=  (/) )
1710, 16sylan 457 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    i^i cin 3151    C_ wss 3152   (/)c0 3455   U.cuni 3827   ` cfv 5255   Topctop 16631   clsccl 16755
This theorem is referenced by:  neindisj  16854  clscon  17156  txcls  17299  ptclsg  17309  flimsncls  17681  hauspwpwf1  17682  met2ndci  18068  metdseq0  18358  heibor1lem  26533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-cld 16756  df-ntr 16757  df-cls 16758
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