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Theorem fbncp 20630
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 20622 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  -.  (/)  e.  F
)
21adantr 463 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  -.  (/) 
e.  F )
3 fbasssin 20627 . . . 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 3843 . . . . . . . 8  |-  ( A  i^i  ( B  \  A ) )  =  (/)
54sseq2i 3466 . . . . . . 7  |-  ( x 
C_  ( A  i^i  ( B  \  A ) )  <->  x  C_  (/) )
6 ss0 3769 . . . . . . 7  |-  ( x 
C_  (/)  ->  x  =  (/) )
75, 6sylbi 195 . . . . . 6  |-  ( x 
C_  ( A  i^i  ( B  \  A ) )  ->  x  =  (/) )
8 eleq1 2474 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  F  <->  (/)  e.  F
) )
98biimpac 484 . . . . . 6  |-  ( ( x  e.  F  /\  x  =  (/) )  ->  (/) 
e.  F )
107, 9sylan2 472 . . . . 5  |-  ( ( x  e.  F  /\  x  C_  ( A  i^i  ( B  \  A ) ) )  ->  (/)  e.  F
)
1110rexlimiva 2891 . . . 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 1199 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   E.wrex 2754    \ cdif 3410    i^i cin 3412    C_ wss 3413   (/)c0 3737   ` cfv 5568   fBascfbas 18724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fv 5576  df-fbas 18734
This theorem is referenced by:  filcon  20674  fgtr  20681  ufilb  20697  ufilmax  20698  ufilen  20721  flimrest  20774  fclsrest  20815  cfilres  22025  relcmpcmet  22045
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