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Theorem neindisj 18839
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
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 neips.1 . . . . . . . 8  |-  X  = 
U. J
21clsss3 18781 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
32sseld 3455 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  ->  P  e.  X ) )
43impr 619 . . . . 5  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) ) )  ->  P  e.  X
)
51isneip 18827 . . . . 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 470 . . . 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 969 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  <->  ( J  e. 
Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) ) )
81clsndisj 18797 . . . . . . . . . . . 12  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) )  /\  ( g  e.  J  /\  P  e.  g
) )  ->  (
g  i^i  S )  =/=  (/) )
97, 8sylanbr 473 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  (
( cls `  J
) `  S )
) )  /\  (
g  e.  J  /\  P  e.  g )
)  ->  ( g  i^i  S )  =/=  (/) )
109anassrs 648 . . . . . . . . . 10  |-  ( ( ( ( J  e. 
Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) )  /\  g  e.  J )  /\  P  e.  g )  ->  (
g  i^i  S )  =/=  (/) )
1110adantllr 718 . . . . . . . . 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 716 . . . . . . . 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 3828 . . . . . . . . . . 11  |-  ( ( g  C_  N  /\  ( N  i^i  S )  =  (/) )  ->  (
g  i^i  S )  =  (/) )
1413ex 434 . . . . . . . . . 10  |-  ( g 
C_  N  ->  (
( N  i^i  S
)  =  (/)  ->  (
g  i^i  S )  =  (/) ) )
1514necon3d 2672 . . . . . . . . 9  |-  ( g 
C_  N  ->  (
( g  i^i  S
)  =/=  (/)  ->  ( N  i^i  S )  =/=  (/) ) )
1615ad2antll 728 . . . . . . . 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 15 . . . . . . 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 434 . . . . . 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 2939 . . . . 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 603 . . . 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 215 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) ) )  ->  ( N  e.  ( ( nei `  J
) `  { P } )  ->  ( N  i^i  S )  =/=  (/) ) )
2221exp32 605 . 2  |-  ( J  e.  Top  ->  ( S  C_  X  ->  ( P  e.  ( ( cls `  J ) `  S )  ->  ( N  e.  ( ( nei `  J ) `  { P } )  -> 
( N  i^i  S
)  =/=  (/) ) ) ) )
2322imp43 595 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796    i^i cin 3427    C_ wss 3428   (/)c0 3737   {csn 3977   U.cuni 4191   ` cfv 5518   Topctop 18616   clsccl 18740   neicnei 18819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-top 18621  df-cld 18741  df-ntr 18742  df-cls 18743  df-nei 18820
This theorem is referenced by:  clslp  18870  flimclslem  19675  utop3cls  19944
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