MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fbncp Structured version   Unicode version

Theorem fbncp 17871
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 17863 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  -.  (/)  e.  F
)
21adantr 452 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  -.  (/) 
e.  F )
3 fbasssin 17868 . . . 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 3700 . . . . . . . 8  |-  ( A  i^i  ( B  \  A ) )  =  (/)
54sseq2i 3373 . . . . . . 7  |-  ( x 
C_  ( A  i^i  ( B  \  A ) )  <->  x  C_  (/) )
6 ss0 3658 . . . . . . 7  |-  ( x 
C_  (/)  ->  x  =  (/) )
75, 6sylbi 188 . . . . . 6  |-  ( x 
C_  ( A  i^i  ( B  \  A ) )  ->  x  =  (/) )
8 eleq1 2496 . . . . . . 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 2825 . . . 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 2706    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   ` cfv 5454   fBascfbas 16689
This theorem is referenced by:  filcon  17915  fgtr  17922  ufilb  17938  ufilmax  17939  ufilen  17962  flimrest  18015  fclsrest  18056  cfilres  19249  relcmpcmet  19269
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-fbas 16699
  Copyright terms: Public domain W3C validator