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Theorem fbasfip 20961
Description: A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
fbasfip  |-  ( F  e.  ( fBas `  X
)  ->  -.  (/)  e.  ( fi `  F ) )

Proof of Theorem fbasfip
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3608 . . . . . 6  |-  ( y  e.  ( ~P F  i^i  Fin )  <->  ( y  e.  ~P F  /\  y  e.  Fin ) )
2 elpwi 3951 . . . . . . 7  |-  ( y  e.  ~P F  -> 
y  C_  F )
32anim1i 578 . . . . . 6  |-  ( ( y  e.  ~P F  /\  y  e.  Fin )  ->  ( y  C_  F  /\  y  e.  Fin ) )
41, 3sylbi 200 . . . . 5  |-  ( y  e.  ( ~P F  i^i  Fin )  ->  (
y  C_  F  /\  y  e.  Fin )
)
5 fbssint 20931 . . . . . 6  |-  ( ( F  e.  ( fBas `  X )  /\  y  C_  F  /\  y  e. 
Fin )  ->  E. z  e.  F  z  C_  |^| y )
653expb 1232 . . . . 5  |-  ( ( F  e.  ( fBas `  X )  /\  (
y  C_  F  /\  y  e.  Fin )
)  ->  E. z  e.  F  z  C_  |^| y )
74, 6sylan2 482 . . . 4  |-  ( ( F  e.  ( fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin ) )  ->  E. z  e.  F  z  C_  |^| y )
8 0nelfb 20924 . . . . . . . . 9  |-  ( F  e.  ( fBas `  X
)  ->  -.  (/)  e.  F
)
98ad2antrr 740 . . . . . . . 8  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  z  e.  F )  ->  -.  (/) 
e.  F )
10 eleq1 2537 . . . . . . . . . 10  |-  ( z  =  (/)  ->  ( z  e.  F  <->  (/)  e.  F
) )
1110biimpcd 232 . . . . . . . . 9  |-  ( z  e.  F  ->  (
z  =  (/)  ->  (/)  e.  F
) )
1211adantl 473 . . . . . . . 8  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  z  e.  F )  ->  (
z  =  (/)  ->  (/)  e.  F
) )
139, 12mtod 182 . . . . . . 7  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  z  e.  F )  ->  -.  z  =  (/) )
14 ss0 3768 . . . . . . 7  |-  ( z 
C_  (/)  ->  z  =  (/) )
1513, 14nsyl 125 . . . . . 6  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  z  e.  F )  ->  -.  z  C_  (/) )
1615adantrr 731 . . . . 5  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  ( z  e.  F  /\  z  C_ 
|^| y ) )  ->  -.  z  C_  (/) )
17 sseq2 3440 . . . . . . 7  |-  ( (/)  =  |^| y  ->  (
z  C_  (/)  <->  z  C_  |^| y ) )
1817biimprcd 233 . . . . . 6  |-  ( z 
C_  |^| y  ->  ( (/)  =  |^| y  -> 
z  C_  (/) ) )
1918ad2antll 743 . . . . 5  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  ( z  e.  F  /\  z  C_ 
|^| y ) )  ->  ( (/)  =  |^| y  ->  z  C_  (/) ) )
2016, 19mtod 182 . . . 4  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  ( z  e.  F  /\  z  C_ 
|^| y ) )  ->  -.  (/)  =  |^| y )
217, 20rexlimddv 2875 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin ) )  ->  -.  (/)  =  |^| y )
2221nrexdv 2842 . 2  |-  ( F  e.  ( fBas `  X
)  ->  -.  E. y  e.  ( ~P F  i^i  Fin ) (/)  =  |^| y )
23 0ex 4528 . . 3  |-  (/)  e.  _V
24 elfi 7945 . . 3  |-  ( (
(/)  e.  _V  /\  F  e.  ( fBas `  X
) )  ->  ( (/) 
e.  ( fi `  F )  <->  E. y  e.  ( ~P F  i^i  Fin ) (/)  =  |^| y ) )
2523, 24mpan 684 . 2  |-  ( F  e.  ( fBas `  X
)  ->  ( (/)  e.  ( fi `  F )  <->  E. y  e.  ( ~P F  i^i  Fin ) (/)  =  |^| y ) )
2622, 25mtbird 308 1  |-  ( F  e.  ( fBas `  X
)  ->  -.  (/)  e.  ( fi `  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031    i^i cin 3389    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   |^|cint 4226   ` cfv 5589   Fincfn 7587   ficfi 7942   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  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  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-reu 2763  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-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  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-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-fin 7591  df-fi 7943  df-fbas 19044
This theorem is referenced by:  fbunfip  20962
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