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Theorem neindisj 19785
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 19727 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
32sseld 3488 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  ->  P  e.  X ) )
43impr 617 . . . . 5  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) ) )  ->  P  e.  X
)
51isneip 19773 . . . . 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 468 . . . 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 975 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  <->  ( J  e. 
Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) ) )
81clsndisj 19743 . . . . . . . . . . . 12  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) )  /\  ( g  e.  J  /\  P  e.  g
) )  ->  (
g  i^i  S )  =/=  (/) )
97, 8sylanbr 471 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  (
( cls `  J
) `  S )
) )  /\  (
g  e.  J  /\  P  e.  g )
)  ->  ( g  i^i  S )  =/=  (/) )
109anassrs 646 . . . . . . . . . 10  |-  ( ( ( ( J  e. 
Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) )  /\  g  e.  J )  /\  P  e.  g )  ->  (
g  i^i  S )  =/=  (/) )
1110adantllr 716 . . . . . . . . 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 714 . . . . . . . 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 3864 . . . . . . . . . . 11  |-  ( ( g  C_  N  /\  ( N  i^i  S )  =  (/) )  ->  (
g  i^i  S )  =  (/) )
1413ex 432 . . . . . . . . . 10  |-  ( g 
C_  N  ->  (
( N  i^i  S
)  =  (/)  ->  (
g  i^i  S )  =  (/) ) )
1514necon3d 2678 . . . . . . . . 9  |-  ( g 
C_  N  ->  (
( g  i^i  S
)  =/=  (/)  ->  ( N  i^i  S )  =/=  (/) ) )
1615ad2antll 726 . . . . . . . 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 432 . . . . . 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 2946 . . . . 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 601 . . . 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 603 . 2  |-  ( J  e.  Top  ->  ( S  C_  X  ->  ( P  e.  ( ( cls `  J ) `  S )  ->  ( N  e.  ( ( nei `  J ) `  { P } )  -> 
( N  i^i  S
)  =/=  (/) ) ) ) )
2322imp43 593 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016   U.cuni 4235   ` cfv 5570   Topctop 19561   clsccl 19686   neicnei 19765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-top 19566  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766
This theorem is referenced by:  clslp  19816  flimclslem  20651  utop3cls  20920
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