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Theorem 0ntr 19338
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 3885 . . . . 5  |-  ( X 
C_  S  <->  ( X  \  S )  =  (/) )
2 eqss 3519 . . . . . . . . 9  |-  ( S  =  X  <->  ( S  C_  X  /\  X  C_  S ) )
3 fveq2 5864 . . . . . . . . . . . . 13  |-  ( S  =  X  ->  (
( int `  J
) `  S )  =  ( ( int `  J ) `  X
) )
4 clscld.1 . . . . . . . . . . . . . 14  |-  X  = 
U. J
54ntrtop 19337 . . . . . . . . . . . . 13  |-  ( J  e.  Top  ->  (
( int `  J
) `  X )  =  X )
63, 5sylan9eqr 2530 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  S  =  X )  ->  ( ( int `  J
) `  S )  =  X )
76eqeq1d 2469 . . . . . . . . . . 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 2691 . . 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 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473    C_ wss 3476   (/)c0 3785   U.cuni 4245   ` cfv 5586   Topctop 19161   intcnt 19284
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  ax-un 6574
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-top 19166  df-ntr 19287
This theorem is referenced by: (None)
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