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Theorem fbssint 17533
Description: A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
fbssint  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  E. x  e.  F  x  C_  |^| A
)
Distinct variable groups:    x, A    x, F    x, B

Proof of Theorem fbssint
StepHypRef Expression
1 fbasne0 17525 . . . . . 6  |-  ( F  e.  ( fBas `  B
)  ->  F  =/=  (/) )
2 n0 3464 . . . . . 6  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
31, 2sylib 188 . . . . 5  |-  ( F  e.  ( fBas `  B
)  ->  E. x  x  e.  F )
4 ssv 3198 . . . . . . . 8  |-  x  C_  _V
54jctr 526 . . . . . . 7  |-  ( x  e.  F  ->  (
x  e.  F  /\  x  C_  _V ) )
65eximi 1563 . . . . . 6  |-  ( E. x  x  e.  F  ->  E. x ( x  e.  F  /\  x  C_ 
_V ) )
7 df-rex 2549 . . . . . 6  |-  ( E. x  e.  F  x 
C_  _V  <->  E. x ( x  e.  F  /\  x  C_ 
_V ) )
86, 7sylibr 203 . . . . 5  |-  ( E. x  x  e.  F  ->  E. x  e.  F  x  C_  _V )
93, 8syl 15 . . . 4  |-  ( F  e.  ( fBas `  B
)  ->  E. x  e.  F  x  C_  _V )
10 inteq 3865 . . . . . . 7  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
11 int0 3876 . . . . . . 7  |-  |^| (/)  =  _V
1210, 11syl6eq 2331 . . . . . 6  |-  ( A  =  (/)  ->  |^| A  =  _V )
1312sseq2d 3206 . . . . 5  |-  ( A  =  (/)  ->  ( x 
C_  |^| A  <->  x  C_  _V ) )
1413rexbidv 2564 . . . 4  |-  ( A  =  (/)  ->  ( E. x  e.  F  x 
C_  |^| A  <->  E. x  e.  F  x  C_  _V ) )
159, 14syl5ibrcom 213 . . 3  |-  ( F  e.  ( fBas `  B
)  ->  ( A  =  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
16153ad2ant1 976 . 2  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  ( A  =  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
17 simpl1 958 . . . 4  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  F  e.  ( fBas `  B ) )
18 simpl2 959 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  C_  F )
19 simpr 447 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  =/=  (/) )
20 simpl3 960 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  e.  Fin )
21 elfir 7169 . . . . 5  |-  ( ( F  e.  ( fBas `  B )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  e.  ( fi
`  F ) )
2217, 18, 19, 20, 21syl13anc 1184 . . . 4  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  |^| A  e.  ( fi
`  F ) )
23 fbssfi 17532 . . . 4  |-  ( ( F  e.  ( fBas `  B )  /\  |^| A  e.  ( fi `  F ) )  ->  E. x  e.  F  x  C_  |^| A )
2417, 22, 23syl2anc 642 . . 3  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  E. x  e.  F  x  C_  |^| A )
2524ex 423 . 2  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  ( A  =/=  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
2616, 25pm2.61dne 2523 1  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  E. x  e.  F  x  C_  |^| A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   _Vcvv 2788    C_ wss 3152   (/)c0 3455   |^|cint 3862   ` cfv 5255   Fincfn 6863   ficfi 7164   fBascfbas 17518
This theorem is referenced by:  fbasfip  17563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-fbas 17520
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