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Theorem fbasfip 19400
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 3536 . . . . . 6  |-  ( y  e.  ( ~P F  i^i  Fin )  <->  ( y  e.  ~P F  /\  y  e.  Fin ) )
2 elpwi 3866 . . . . . . 7  |-  ( y  e.  ~P F  -> 
y  C_  F )
32anim1i 565 . . . . . 6  |-  ( ( y  e.  ~P F  /\  y  e.  Fin )  ->  ( y  C_  F  /\  y  e.  Fin ) )
41, 3sylbi 195 . . . . 5  |-  ( y  e.  ( ~P F  i^i  Fin )  ->  (
y  C_  F  /\  y  e.  Fin )
)
5 fbssint 19370 . . . . . 6  |-  ( ( F  e.  ( fBas `  X )  /\  y  C_  F  /\  y  e. 
Fin )  ->  E. z  e.  F  z  C_  |^| y )
653expb 1183 . . . . 5  |-  ( ( F  e.  ( fBas `  X )  /\  (
y  C_  F  /\  y  e.  Fin )
)  ->  E. z  e.  F  z  C_  |^| y )
74, 6sylan2 471 . . . 4  |-  ( ( F  e.  ( fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin ) )  ->  E. z  e.  F  z  C_  |^| y )
8 0nelfb 19363 . . . . . . . . 9  |-  ( F  e.  ( fBas `  X
)  ->  -.  (/)  e.  F
)
98ad2antrr 720 . . . . . . . 8  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  z  e.  F )  ->  -.  (/) 
e.  F )
10 eleq1 2501 . . . . . . . . . 10  |-  ( z  =  (/)  ->  ( z  e.  F  <->  (/)  e.  F
) )
1110biimpcd 224 . . . . . . . . 9  |-  ( z  e.  F  ->  (
z  =  (/)  ->  (/)  e.  F
) )
1211adantl 463 . . . . . . . 8  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  z  e.  F )  ->  (
z  =  (/)  ->  (/)  e.  F
) )
139, 12mtod 177 . . . . . . 7  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  z  e.  F )  ->  -.  z  =  (/) )
14 ss0 3665 . . . . . . 7  |-  ( z 
C_  (/)  ->  z  =  (/) )
1513, 14nsyl 121 . . . . . 6  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  z  e.  F )  ->  -.  z  C_  (/) )
1615adantrr 711 . . . . 5  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  ( z  e.  F  /\  z  C_ 
|^| y ) )  ->  -.  z  C_  (/) )
17 sseq2 3375 . . . . . . 7  |-  ( (/)  =  |^| y  ->  (
z  C_  (/)  <->  z  C_  |^| y ) )
1817biimprcd 225 . . . . . 6  |-  ( z 
C_  |^| y  ->  ( (/)  =  |^| y  -> 
z  C_  (/) ) )
1918ad2antll 723 . . . . 5  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  ( z  e.  F  /\  z  C_ 
|^| y ) )  ->  ( (/)  =  |^| y  ->  z  C_  (/) ) )
2016, 19mtod 177 . . . 4  |-  ( ( ( F  e.  (
fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin )
)  /\  ( z  e.  F  /\  z  C_ 
|^| y ) )  ->  -.  (/)  =  |^| y )
217, 20rexlimddv 2843 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  y  e.  ( ~P F  i^i  Fin ) )  ->  -.  (/)  =  |^| y )
2221nrexdv 2817 . 2  |-  ( F  e.  ( fBas `  X
)  ->  -.  E. y  e.  ( ~P F  i^i  Fin ) (/)  =  |^| y )
23 0ex 4419 . . 3  |-  (/)  e.  _V
24 elfi 7659 . . 3  |-  ( (
(/)  e.  _V  /\  F  e.  ( fBas `  X
) )  ->  ( (/) 
e.  ( fi `  F )  <->  E. y  e.  ( ~P F  i^i  Fin ) (/)  =  |^| y ) )
2523, 24mpan 665 . 2  |-  ( F  e.  ( fBas `  X
)  ->  ( (/)  e.  ( fi `  F )  <->  E. y  e.  ( ~P F  i^i  Fin ) (/)  =  |^| y ) )
2622, 25mtbird 301 1  |-  ( F  e.  ( fBas `  X
)  ->  -.  (/)  e.  ( fi `  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   E.wrex 2714   _Vcvv 2970    i^i cin 3324    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   |^|cint 4125   ` cfv 5415   Fincfn 7306   ficfi 7656   fBascfbas 17763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-fin 7310  df-fi 7657  df-fbas 17773
This theorem is referenced by:  fbunfip  19401
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