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Theorem clsndisj 16644
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 16628 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
54sseld 3102 . . . 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 16642 . . . 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 2314 . . . . 5  |-  ( x  =  U  ->  ( P  e.  x  <->  P  e.  U ) )
12 ineq1 3271 . . . . . 6  |-  ( x  =  U  ->  (
x  i^i  S )  =  ( U  i^i  S ) )
1312neeq1d 2425 . . . . 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 )  =/=  (/) ) ) )
1514rcla4cv 2818 . . 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 1619    e. wcel 1621    =/= wne 2412   A.wral 2509    i^i cin 3077    C_ wss 3078   (/)c0 3362   U.cuni 3727   ` cfv 4592   Topctop 16463   clsccl 16587
This theorem is referenced by:  neindisj  16686  clscon  16988  txcls  17131  ptclsg  17141  flimsncls  17513  hauspwpwf1  17514  met2ndci  17900  metdseq0  18190  heibor1lem  25699
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-top 16468  df-cld 16588  df-ntr 16589  df-cls 16590
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