MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssntr Structured version   Unicode version

Theorem ssntr 18637
Description: An open subset of a set is a subset of the set's interior. (Contributed by Jeff Hankins, 31-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
ssntr  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S ) )  ->  O  C_  (
( int `  J
) `  S )
)

Proof of Theorem ssntr
StepHypRef Expression
1 elin 3534 . . . . 5  |-  ( O  e.  ( J  i^i  ~P S )  <->  ( O  e.  J  /\  O  e. 
~P S ) )
2 elpwg 3863 . . . . . 6  |-  ( O  e.  J  ->  ( O  e.  ~P S  <->  O 
C_  S ) )
32pm5.32i 637 . . . . 5  |-  ( ( O  e.  J  /\  O  e.  ~P S
)  <->  ( O  e.  J  /\  O  C_  S ) )
41, 3bitr2i 250 . . . 4  |-  ( ( O  e.  J  /\  O  C_  S )  <->  O  e.  ( J  i^i  ~P S
) )
5 elssuni 4116 . . . 4  |-  ( O  e.  ( J  i^i  ~P S )  ->  O  C_ 
U. ( J  i^i  ~P S ) )
64, 5sylbi 195 . . 3  |-  ( ( O  e.  J  /\  O  C_  S )  ->  O  C_  U. ( J  i^i  ~P S ) )
76adantl 466 . 2  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S ) )  ->  O  C_  U. ( J  i^i  ~P S ) )
8 clscld.1 . . . 4  |-  X  = 
U. J
98ntrval 18615 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
109adantr 465 . 2  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S ) )  ->  ( ( int `  J ) `  S )  =  U. ( J  i^i  ~P S
) )
117, 10sseqtr4d 3388 1  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S ) )  ->  O  C_  (
( int `  J
) `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3322    C_ wss 3323   ~Pcpw 3855   U.cuni 4086   ` cfv 5413   Topctop 18473   intcnt 18596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-top 18478  df-ntr 18599
This theorem is referenced by:  ntrin  18640  neiint  18683  restntr  18761  cnntri  18850  xkococnlem  19207  iccntr  20373  bcthlem5  20814  ftc1  21489  lgamucov  26976  cvmlift2lem12  27155  cvmlift3lem7  27166  ftc1cnnc  28419  opnregcld  28478
  Copyright terms: Public domain W3C validator