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Theorem fbssint 20464
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 20456 . . . . . 6  |-  ( F  e.  ( fBas `  B
)  ->  F  =/=  (/) )
2 n0 3803 . . . . . 6  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
31, 2sylib 196 . . . . 5  |-  ( F  e.  ( fBas `  B
)  ->  E. x  x  e.  F )
4 ssv 3519 . . . . . . . 8  |-  x  C_  _V
54jctr 542 . . . . . . 7  |-  ( x  e.  F  ->  (
x  e.  F  /\  x  C_  _V ) )
65eximi 1657 . . . . . 6  |-  ( E. x  x  e.  F  ->  E. x ( x  e.  F  /\  x  C_ 
_V ) )
7 df-rex 2813 . . . . . 6  |-  ( E. x  e.  F  x 
C_  _V  <->  E. x ( x  e.  F  /\  x  C_ 
_V ) )
86, 7sylibr 212 . . . . 5  |-  ( E. x  x  e.  F  ->  E. x  e.  F  x  C_  _V )
93, 8syl 16 . . . 4  |-  ( F  e.  ( fBas `  B
)  ->  E. x  e.  F  x  C_  _V )
10 inteq 4291 . . . . . . 7  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
11 int0 4302 . . . . . . 7  |-  |^| (/)  =  _V
1210, 11syl6eq 2514 . . . . . 6  |-  ( A  =  (/)  ->  |^| A  =  _V )
1312sseq2d 3527 . . . . 5  |-  ( A  =  (/)  ->  ( x 
C_  |^| A  <->  x  C_  _V ) )
1413rexbidv 2968 . . . 4  |-  ( A  =  (/)  ->  ( E. x  e.  F  x 
C_  |^| A  <->  E. x  e.  F  x  C_  _V ) )
159, 14syl5ibrcom 222 . . 3  |-  ( F  e.  ( fBas `  B
)  ->  ( A  =  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
16153ad2ant1 1017 . 2  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  ( A  =  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
17 simpl1 999 . . . 4  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  F  e.  ( fBas `  B ) )
18 simpl2 1000 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  C_  F )
19 simpr 461 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  =/=  (/) )
20 simpl3 1001 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  e.  Fin )
21 elfir 7893 . . . . 5  |-  ( ( F  e.  ( fBas `  B )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  e.  ( fi
`  F ) )
2217, 18, 19, 20, 21syl13anc 1230 . . . 4  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  |^| A  e.  ( fi
`  F ) )
23 fbssfi 20463 . . . 4  |-  ( ( F  e.  ( fBas `  B )  /\  |^| A  e.  ( fi `  F ) )  ->  E. x  e.  F  x  C_  |^| A )
2417, 22, 23syl2anc 661 . . 3  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  E. x  e.  F  x  C_  |^| A )
2524ex 434 . 2  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  ( A  =/=  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
2616, 25pm2.61dne 2774 1  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  E. x  e.  F  x  C_  |^| A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   E.wrex 2808   _Vcvv 3109    C_ wss 3471   (/)c0 3793   |^|cint 4288   ` cfv 5594   Fincfn 7535   ficfi 7888   fBascfbas 18532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-fin 7539  df-fi 7889  df-fbas 18542
This theorem is referenced by:  fbasfip  20494
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