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Theorem fbncp 20075
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 20067 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  -.  (/)  e.  F
)
21adantr 465 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  -.  (/) 
e.  F )
3 fbasssin 20072 . . . 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 3899 . . . . . . . 8  |-  ( A  i^i  ( B  \  A ) )  =  (/)
54sseq2i 3529 . . . . . . 7  |-  ( x 
C_  ( A  i^i  ( B  \  A ) )  <->  x  C_  (/) )
6 ss0 3816 . . . . . . 7  |-  ( x 
C_  (/)  ->  x  =  (/) )
75, 6sylbi 195 . . . . . 6  |-  ( x 
C_  ( A  i^i  ( B  \  A ) )  ->  x  =  (/) )
8 eleq1 2539 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  F  <->  (/)  e.  F
) )
98biimpac 486 . . . . . 6  |-  ( ( x  e.  F  /\  x  =  (/) )  ->  (/) 
e.  F )
107, 9sylan2 474 . . . . 5  |-  ( ( x  e.  F  /\  x  C_  ( A  i^i  ( B  \  A ) ) )  ->  (/)  e.  F
)
1110rexlimiva 2951 . . . 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 1198 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  (
( B  \  A
)  e.  F  ->  (/) 
e.  F ) )
142, 13mtod 177 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   ` cfv 5586   fBascfbas 18177
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
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-nel 2665  df-ral 2819  df-rex 2820  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-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-fv 5594  df-fbas 18187
This theorem is referenced by:  filcon  20119  fgtr  20126  ufilb  20142  ufilmax  20143  ufilen  20166  flimrest  20219  fclsrest  20260  cfilres  21470  relcmpcmet  21490
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