ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  topssnei Unicode version

Theorem topssnei 12029
Description: A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)
Hypotheses
Ref Expression
tpnei.1  |-  X  = 
U. J
topssnei.2  |-  Y  = 
U. K
Assertion
Ref Expression
topssnei  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  J  C_  K )  ->  ( ( nei `  J ) `  S
)  C_  ( ( nei `  K ) `  S ) )

Proof of Theorem topssnei
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl2 950 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  K  e.  Top )
2 simprl 499 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  J  C_  K )
3 simpl1 949 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  J  e.  Top )
4 simprr 500 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  e.  ( ( nei `  J
) `  S )
)
5 tpnei.1 . . . . . . . . 9  |-  X  = 
U. J
65neii1 12014 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S ) )  ->  x  C_  X )
73, 4, 6syl2anc 404 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  C_  X )
85ntropn 11984 . . . . . . 7  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( int `  J
) `  x )  e.  J )
93, 7, 8syl2anc 404 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  e.  J )
102, 9sseldd 3040 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  e.  K )
115neiss2 12009 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
123, 4, 11syl2anc 404 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  S  C_  X )
135neiint 12012 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  X  /\  x  C_  X )  ->  (
x  e.  ( ( nei `  J ) `
 S )  <->  S  C_  (
( int `  J
) `  x )
) )
143, 12, 7, 13syl3anc 1181 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
x  e.  ( ( nei `  J ) `
 S )  <->  S  C_  (
( int `  J
) `  x )
) )
154, 14mpbid 146 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  S  C_  ( ( int `  J
) `  x )
)
16 opnneiss 12025 . . . . 5  |-  ( ( K  e.  Top  /\  ( ( int `  J
) `  x )  e.  K  /\  S  C_  ( ( int `  J
) `  x )
)  ->  ( ( int `  J ) `  x )  e.  ( ( nei `  K
) `  S )
)
171, 10, 15, 16syl3anc 1181 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  e.  ( ( nei `  K
) `  S )
)
185ntrss2 11988 . . . . 5  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( int `  J
) `  x )  C_  x )
193, 7, 18syl2anc 404 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  C_  x )
20 simpl3 951 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  X  =  Y )
217, 20sseqtrd 3077 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  C_  Y )
22 topssnei.2 . . . . 5  |-  Y  = 
U. K
2322ssnei2 12024 . . . 4  |-  ( ( ( K  e.  Top  /\  ( ( int `  J
) `  x )  e.  ( ( nei `  K
) `  S )
)  /\  ( (
( int `  J
) `  x )  C_  x  /\  x  C_  Y ) )  ->  x  e.  ( ( nei `  K ) `  S ) )
241, 17, 19, 21, 23syl22anc 1182 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  e.  ( ( nei `  K
) `  S )
)
2524expr 368 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  J  C_  K )  ->  ( x  e.  ( ( nei `  J
) `  S )  ->  x  e.  ( ( nei `  K ) `
 S ) ) )
2625ssrdv 3045 1  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  J  C_  K )  ->  ( ( nei `  J ) `  S
)  C_  ( ( nei `  K ) `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 927    = wceq 1296    e. wcel 1445    C_ wss 3013   U.cuni 3675   ` cfv 5049   Topctop 11863   intcnt 11960   neicnei 12005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-top 11864  df-ntr 11963  df-nei 12006
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator