MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0ntr Structured version   Unicode version

Theorem 0ntr 18802
Description: A subset with an empty interior cannot cover a whole (nonempty) topology. (Contributed by NM, 12-Sep-2006.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
0ntr  |-  ( ( ( J  e.  Top  /\  X  =/=  (/) )  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( X  \  S
)  =/=  (/) )

Proof of Theorem 0ntr
StepHypRef Expression
1 ssdif0 3840 . . . . 5  |-  ( X 
C_  S  <->  ( X  \  S )  =  (/) )
2 eqss 3474 . . . . . . . . 9  |-  ( S  =  X  <->  ( S  C_  X  /\  X  C_  S ) )
3 fveq2 5794 . . . . . . . . . . . . 13  |-  ( S  =  X  ->  (
( int `  J
) `  S )  =  ( ( int `  J ) `  X
) )
4 clscld.1 . . . . . . . . . . . . . 14  |-  X  = 
U. J
54ntrtop 18801 . . . . . . . . . . . . 13  |-  ( J  e.  Top  ->  (
( int `  J
) `  X )  =  X )
63, 5sylan9eqr 2515 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  S  =  X )  ->  ( ( int `  J
) `  S )  =  X )
76eqeq1d 2454 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  S  =  X )  ->  ( ( ( int `  J ) `  S
)  =  (/)  <->  X  =  (/) ) )
87biimpd 207 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  S  =  X )  ->  ( ( ( int `  J ) `  S
)  =  (/)  ->  X  =  (/) ) )
98ex 434 . . . . . . . . 9  |-  ( J  e.  Top  ->  ( S  =  X  ->  ( ( ( int `  J
) `  S )  =  (/)  ->  X  =  (/) ) ) )
102, 9syl5bir 218 . . . . . . . 8  |-  ( J  e.  Top  ->  (
( S  C_  X  /\  X  C_  S )  ->  ( ( ( int `  J ) `
 S )  =  (/)  ->  X  =  (/) ) ) )
1110expd 436 . . . . . . 7  |-  ( J  e.  Top  ->  ( S  C_  X  ->  ( X  C_  S  ->  (
( ( int `  J
) `  S )  =  (/)  ->  X  =  (/) ) ) ) )
1211com34 83 . . . . . 6  |-  ( J  e.  Top  ->  ( S  C_  X  ->  (
( ( int `  J
) `  S )  =  (/)  ->  ( X  C_  S  ->  X  =  (/) ) ) ) )
1312imp32 433 . . . . 5  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( X  C_  S  ->  X  =  (/) ) )
141, 13syl5bir 218 . . . 4  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( ( X  \  S )  =  (/)  ->  X  =  (/) ) )
1514necon3d 2673 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( X  =/=  (/)  ->  ( X  \  S )  =/=  (/) ) )
1615imp 429 . 2  |-  ( ( ( J  e.  Top  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  /\  X  =/=  (/) )  ->  ( X  \  S )  =/=  (/) )
1716an32s 802 1  |-  ( ( ( J  e.  Top  /\  X  =/=  (/) )  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( X  \  S
)  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645    \ cdif 3428    C_ wss 3431   (/)c0 3740   U.cuni 4194   ` cfv 5521   Topctop 18625   intcnt 18748
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-top 18630  df-ntr 18751
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator