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Theorem fbncp 17863
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 17855 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  -.  (/)  e.  F
)
21adantr 452 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  -.  (/) 
e.  F )
3 fbasssin 17860 . . . 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 3692 . . . . . . . 8  |-  ( A  i^i  ( B  \  A ) )  =  (/)
54sseq2i 3365 . . . . . . 7  |-  ( x 
C_  ( A  i^i  ( B  \  A ) )  <->  x  C_  (/) )
6 ss0 3650 . . . . . . 7  |-  ( x 
C_  (/)  ->  x  =  (/) )
75, 6sylbi 188 . . . . . 6  |-  ( x 
C_  ( A  i^i  ( B  \  A ) )  ->  x  =  (/) )
8 eleq1 2495 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  F  <->  (/)  e.  F
) )
98biimpac 473 . . . . . 6  |-  ( ( x  e.  F  /\  x  =  (/) )  ->  (/) 
e.  F )
107, 9sylan2 461 . . . . 5  |-  ( ( x  e.  F  /\  x  C_  ( A  i^i  ( B  \  A ) ) )  ->  (/)  e.  F
)
1110rexlimiva 2817 . . . 4  |-  ( E. x  e.  F  x 
C_  ( A  i^i  ( B  \  A ) )  ->  (/)  e.  F
)
123, 11syl 16 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F  /\  ( B  \  A )  e.  F )  ->  (/)  e.  F
)
13123expia 1155 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  (
( B  \  A
)  e.  F  ->  (/) 
e.  F ) )
142, 13mtod 170 1  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  -.  ( B  \  A )  e.  F )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2698    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   ` cfv 5446   fBascfbas 16681
This theorem is referenced by:  filcon  17907  fgtr  17914  ufilb  17930  ufilmax  17931  ufilen  17954  flimrest  18007  fclsrest  18048  cfilres  19241  relcmpcmet  19261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-fbas 16691
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