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Theorem neii1 19401
Description: A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.)
Hypothesis
Ref Expression
neifval.1  |-  X  = 
U. J
Assertion
Ref Expression
neii1  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  N  C_  X )

Proof of Theorem neii1
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . 3  |-  X  = 
U. J
21neiss2 19396 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
31isnei 19398 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
4 simpl 457 . . . 4  |-  ( ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )  ->  N  C_  X
)
53, 4syl6bi 228 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  ->  N  C_  X )
)
65impancom 440 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  -> 
( S  C_  X  ->  N  C_  X )
)
72, 6mpd 15 1  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  N  C_  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815    C_ wss 3476   U.cuni 4245   ` cfv 5588   Topctop 19189   neicnei 19392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-top 19194  df-nei 19393
This theorem is referenced by:  neisspw  19402  neiss  19404  opnnei  19415  neiuni  19417  topssnei  19419  innei  19420  neissex  19422  iscnp4  19558  llycmpkgen2  19814  neitx  19871  flimopn  20239  flfnei  20255  fclsneii  20281  fcfnei  20299  cnextcn  20330  limcflf  22048  cvmlift2lem1  28415  neiin  29755  neibastop2  29810
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