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Theorem 0ntr 18634
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 3734 . . . . 5  |-  ( X 
C_  S  <->  ( X  \  S )  =  (/) )
2 eqss 3368 . . . . . . . . 9  |-  ( S  =  X  <->  ( S  C_  X  /\  X  C_  S ) )
3 fveq2 5688 . . . . . . . . . . . . 13  |-  ( S  =  X  ->  (
( int `  J
) `  S )  =  ( ( int `  J ) `  X
) )
4 clscld.1 . . . . . . . . . . . . . 14  |-  X  = 
U. J
54ntrtop 18633 . . . . . . . . . . . . 13  |-  ( J  e.  Top  ->  (
( int `  J
) `  X )  =  X )
63, 5sylan9eqr 2495 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  S  =  X )  ->  ( ( int `  J
) `  S )  =  X )
76eqeq1d 2449 . . . . . . . . . . 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  =  (/) ) ) )
1110exp3a 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 2644 . . 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 797 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 1364    e. wcel 1761    =/= wne 2604    \ cdif 3322    C_ wss 3325   (/)c0 3634   U.cuni 4088   ` cfv 5415   Topctop 18457   intcnt 18580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-top 18462  df-ntr 18583
This theorem is referenced by: (None)
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