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Theorem fbncp 20932
Description: A filter base does not contain complements of its elements. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
fbncp  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  -.  ( B  \  A )  e.  F )

Proof of Theorem fbncp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 0nelfb 20924 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  -.  (/)  e.  F
)
21adantr 472 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  -.  (/) 
e.  F )
3 fbasssin 20929 . . . 4  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F  /\  ( B  \  A )  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  ( B  \  A ) ) )
4 disjdif 3830 . . . . . . . 8  |-  ( A  i^i  ( B  \  A ) )  =  (/)
54sseq2i 3443 . . . . . . 7  |-  ( x 
C_  ( A  i^i  ( B  \  A ) )  <->  x  C_  (/) )
6 ss0 3768 . . . . . . 7  |-  ( x 
C_  (/)  ->  x  =  (/) )
75, 6sylbi 200 . . . . . 6  |-  ( x 
C_  ( A  i^i  ( B  \  A ) )  ->  x  =  (/) )
8 eleq1 2537 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  F  <->  (/)  e.  F
) )
98biimpac 494 . . . . . 6  |-  ( ( x  e.  F  /\  x  =  (/) )  ->  (/) 
e.  F )
107, 9sylan2 482 . . . . 5  |-  ( ( x  e.  F  /\  x  C_  ( A  i^i  ( B  \  A ) ) )  ->  (/)  e.  F
)
1110rexlimiva 2868 . . . 4  |-  ( E. x  e.  F  x 
C_  ( A  i^i  ( B  \  A ) )  ->  (/)  e.  F
)
123, 11syl 17 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F  /\  ( B  \  A )  e.  F )  ->  (/)  e.  F
)
13123expia 1233 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  (
( B  \  A
)  e.  F  ->  (/) 
e.  F ) )
142, 13mtod 182 1  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  -.  ( B  \  A )  e.  F )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   E.wrex 2757    \ cdif 3387    i^i cin 3389    C_ wss 3390   (/)c0 3722   ` cfv 5589   fBascfbas 19035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-fbas 19044
This theorem is referenced by:  filcon  20976  fgtr  20983  ufilb  20999  ufilmax  21000  ufilen  21023  flimrest  21076  fclsrest  21117  cfilres  22344  relcmpcmet  22364
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