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Theorem neindisj 16686
Description: Any neighborhood of an element in the closure of a subset intersects the subset. Part of proof of Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
Hypothesis
Ref Expression
neips.1  |-  X  = 
U. J
Assertion
Ref Expression
neindisj  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( P  e.  ( ( cls `  J
) `  S )  /\  N  e.  (
( nei `  J
) `  { P } ) ) )  ->  ( N  i^i  S )  =/=  (/) )

Proof of Theorem neindisj
StepHypRef Expression
1 neips.1 . . . . . . . 8  |-  X  = 
U. J
21clsss3 16628 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
32sseld 3102 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  ->  P  e.  X ) )
43impr 605 . . . . 5  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) ) )  ->  P  e.  X
)
51isneip 16674 . . . . 5  |-  ( ( J  e.  Top  /\  P  e.  X )  ->  ( N  e.  ( ( nei `  J
) `  { P } )  <->  ( N  C_  X  /\  E. g  e.  J  ( P  e.  g  /\  g  C_  N ) ) ) )
64, 5syldan 458 . . . 4  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) ) )  ->  ( N  e.  ( ( nei `  J
) `  { P } )  <->  ( N  C_  X  /\  E. g  e.  J  ( P  e.  g  /\  g  C_  N ) ) ) )
7 3anass 943 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  <->  ( J  e. 
Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) ) )
81clsndisj 16644 . . . . . . . . . . . 12  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) )  /\  ( g  e.  J  /\  P  e.  g
) )  ->  (
g  i^i  S )  =/=  (/) )
97, 8sylanbr 461 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  (
( cls `  J
) `  S )
) )  /\  (
g  e.  J  /\  P  e.  g )
)  ->  ( g  i^i  S )  =/=  (/) )
109anassrs 632 . . . . . . . . . 10  |-  ( ( ( ( J  e. 
Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) )  /\  g  e.  J )  /\  P  e.  g )  ->  (
g  i^i  S )  =/=  (/) )
1110adantllr 702 . . . . . . . . 9  |-  ( ( ( ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) )  /\  N  C_  X )  /\  g  e.  J )  /\  P  e.  g )  ->  (
g  i^i  S )  =/=  (/) )
1211adantrr 700 . . . . . . . 8  |-  ( ( ( ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) )  /\  N  C_  X )  /\  g  e.  J )  /\  ( P  e.  g  /\  g  C_  N ) )  ->  ( g  i^i 
S )  =/=  (/) )
13 ssdisj 3411 . . . . . . . . . . 11  |-  ( ( g  C_  N  /\  ( N  i^i  S )  =  (/) )  ->  (
g  i^i  S )  =  (/) )
1413ex 425 . . . . . . . . . 10  |-  ( g 
C_  N  ->  (
( N  i^i  S
)  =  (/)  ->  (
g  i^i  S )  =  (/) ) )
1514necon3d 2450 . . . . . . . . 9  |-  ( g 
C_  N  ->  (
( g  i^i  S
)  =/=  (/)  ->  ( N  i^i  S )  =/=  (/) ) )
1615ad2antll 712 . . . . . . . 8  |-  ( ( ( ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) )  /\  N  C_  X )  /\  g  e.  J )  /\  ( P  e.  g  /\  g  C_  N ) )  ->  ( ( g  i^i  S )  =/=  (/)  ->  ( N  i^i  S )  =/=  (/) ) )
1712, 16mpd 16 . . . . . . 7  |-  ( ( ( ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) )  /\  N  C_  X )  /\  g  e.  J )  /\  ( P  e.  g  /\  g  C_  N ) )  ->  ( N  i^i  S )  =/=  (/) )
1817ex 425 . . . . . 6  |-  ( ( ( ( J  e. 
Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) )  /\  N  C_  X )  /\  g  e.  J )  ->  (
( P  e.  g  /\  g  C_  N
)  ->  ( N  i^i  S )  =/=  (/) ) )
1918rexlimdva 2629 . . . . 5  |-  ( ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  (
( cls `  J
) `  S )
) )  /\  N  C_  X )  ->  ( E. g  e.  J  ( P  e.  g  /\  g  C_  N )  ->  ( N  i^i  S )  =/=  (/) ) )
2019expimpd 589 . . . 4  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) ) )  ->  ( ( N 
C_  X  /\  E. g  e.  J  ( P  e.  g  /\  g  C_  N ) )  ->  ( N  i^i  S )  =/=  (/) ) )
216, 20sylbid 208 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) ) )  ->  ( N  e.  ( ( nei `  J
) `  { P } )  ->  ( N  i^i  S )  =/=  (/) ) )
2221exp32 591 . 2  |-  ( J  e.  Top  ->  ( S  C_  X  ->  ( P  e.  ( ( cls `  J ) `  S )  ->  ( N  e.  ( ( nei `  J ) `  { P } )  -> 
( N  i^i  S
)  =/=  (/) ) ) ) )
2322imp43 581 1  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( P  e.  ( ( cls `  J
) `  S )  /\  N  e.  (
( nei `  J
) `  { P } ) ) )  ->  ( N  i^i  S )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510    i^i cin 3077    C_ wss 3078   (/)c0 3362   {csn 3544   U.cuni 3727   ` cfv 4592   Topctop 16463   clsccl 16587   neicnei 16666
This theorem is referenced by:  clslp  16711  flimclslem  17511  islimrs4  24748
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  df-nei 16667
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