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Theorem efilcp 24920
Description: A filter containing a set  A exists iff  A has the finite intersection property (i.e. no finite intersection of elements of  A is empty). Bourbaki TG I.37 prop. 1. (Contributed by FL, 20-Nov-2007.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
efilcp  |-  ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  -> 
( -.  (/)  e.  ( fi `  A )  <->  E. x  e.  ( Fil `  B ) A 
C_  x ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem efilcp
StepHypRef Expression
1 snfil 17522 . . . . . . . 8  |-  ( ( B  e.  V  /\  B  =/=  (/) )  ->  { B }  e.  ( Fil `  B ) )
213adant1 978 . . . . . . 7  |-  ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  ->  { B }  e.  ( Fil `  B ) )
3 0ss 3458 . . . . . . 7  |-  (/)  C_  { B }
4 sseq2 3175 . . . . . . . 8  |-  ( x  =  { B }  ->  ( (/)  C_  x  <->  (/)  C_  { B } ) )
54rcla4ev 2859 . . . . . . 7  |-  ( ( { B }  e.  ( Fil `  B )  /\  (/)  C_  { B } )  ->  E. x  e.  ( Fil `  B
) (/)  C_  x )
62, 3, 5sylancl 646 . . . . . 6  |-  ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  ->  E. x  e.  ( Fil `  B ) (/)  C_  x )
7 sseq1 3174 . . . . . . 7  |-  ( A  =  (/)  ->  ( A 
C_  x  <->  (/)  C_  x
) )
87rexbidv 2539 . . . . . 6  |-  ( A  =  (/)  ->  ( E. x  e.  ( Fil `  B ) A  C_  x 
<->  E. x  e.  ( Fil `  B )
(/)  C_  x ) )
96, 8syl5ibrcom 215 . . . . 5  |-  ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  -> 
( A  =  (/)  ->  E. x  e.  ( Fil `  B ) A  C_  x )
)
109imp 420 . . . 4  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  A  =  (/) )  ->  E. x  e.  ( Fil `  B ) A 
C_  x )
1110a1d 24 . . 3  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  A  =  (/) )  -> 
( -.  (/)  e.  ( fi `  A )  ->  E. x  e.  ( Fil `  B ) A  C_  x )
)
12 simpl1 963 . . . . . . 7  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  C_  ~P B )
13 simprl 735 . . . . . . 7  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  =/=  (/) )
14 simprr 736 . . . . . . 7  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  -.  (/)  e.  ( fi `  A ) )
15 simpl2 964 . . . . . . . 8  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  B  e.  V
)
16 fsubbas 17525 . . . . . . . 8  |-  ( B  e.  V  ->  (
( fi `  A
)  e.  ( fBas `  B )  <->  ( A  C_ 
~P B  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi `  A ) ) ) )
1715, 16syl 17 . . . . . . 7  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( ( fi
`  A )  e.  ( fBas `  B
)  <->  ( A  C_  ~P B  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi `  A ) ) ) )
1812, 13, 14, 17mpbir3and 1140 . . . . . 6  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( fi `  A )  e.  (
fBas `  B )
)
19 fgcl 17536 . . . . . 6  |-  ( ( fi `  A )  e.  ( fBas `  B
)  ->  ( B filGen ( fi `  A
) )  e.  ( Fil `  B ) )
2018, 19syl 17 . . . . 5  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( B filGen ( fi `  A ) )  e.  ( Fil `  B ) )
21 pwexg 4166 . . . . . . . 8  |-  ( B  e.  V  ->  ~P B  e.  _V )
2215, 21syl 17 . . . . . . 7  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ~P B  e. 
_V )
23 ssexg 4134 . . . . . . . 8  |-  ( ( A  C_  ~P B  /\  ~P B  e.  _V )  ->  A  e.  _V )
24 ssfii 7140 . . . . . . . 8  |-  ( A  e.  _V  ->  A  C_  ( fi `  A
) )
2523, 24syl 17 . . . . . . 7  |-  ( ( A  C_  ~P B  /\  ~P B  e.  _V )  ->  A  C_  ( fi `  A ) )
2612, 22, 25syl2anc 645 . . . . . 6  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  C_  ( fi `  A ) )
27 ssfg 17530 . . . . . . 7  |-  ( ( fi `  A )  e.  ( fBas `  B
)  ->  ( fi `  A )  C_  ( B filGen ( fi `  A ) ) )
2818, 27syl 17 . . . . . 6  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( fi `  A )  C_  ( B filGen ( fi `  A ) ) )
2926, 28sstrd 3164 . . . . 5  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  C_  ( B filGen ( fi `  A ) ) )
30 sseq2 3175 . . . . . 6  |-  ( x  =  ( B filGen ( fi `  A ) )  ->  ( A  C_  x  <->  A  C_  ( B
filGen ( fi `  A
) ) ) )
3130rcla4ev 2859 . . . . 5  |-  ( ( ( B filGen ( fi
`  A ) )  e.  ( Fil `  B
)  /\  A  C_  ( B filGen ( fi `  A ) ) )  ->  E. x  e.  ( Fil `  B ) A  C_  x )
3220, 29, 31syl2anc 645 . . . 4  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  E. x  e.  ( Fil `  B ) A  C_  x )
3332expr 601 . . 3  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  A  =/=  (/) )  ->  ( -.  (/)  e.  ( fi
`  A )  ->  E. x  e.  ( Fil `  B ) A 
C_  x ) )
3411, 33pm2.61dane 2499 . 2  |-  ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  -> 
( -.  (/)  e.  ( fi `  A )  ->  E. x  e.  ( Fil `  B ) A  C_  x )
)
35 filfbas 17506 . . . 4  |-  ( x  e.  ( Fil `  B
)  ->  x  e.  ( fBas `  B )
)
36 fbasfip 17526 . . . 4  |-  ( x  e.  ( fBas `  B
)  ->  -.  (/)  e.  ( fi `  x ) )
37 vex 2766 . . . . . . 7  |-  x  e. 
_V
38 fiss 7145 . . . . . . 7  |-  ( ( x  e.  _V  /\  A  C_  x )  -> 
( fi `  A
)  C_  ( fi `  x ) )
3937, 38mpan 654 . . . . . 6  |-  ( A 
C_  x  ->  ( fi `  A )  C_  ( fi `  x ) )
4039sseld 3154 . . . . 5  |-  ( A 
C_  x  ->  ( (/) 
e.  ( fi `  A )  ->  (/)  e.  ( fi `  x ) ) )
4140con3rr3 130 . . . 4  |-  ( -.  (/)  e.  ( fi `  x )  ->  ( A  C_  x  ->  -.  (/) 
e.  ( fi `  A ) ) )
4235, 36, 413syl 20 . . 3  |-  ( x  e.  ( Fil `  B
)  ->  ( A  C_  x  ->  -.  (/)  e.  ( fi `  A ) ) )
4342rexlimiv 2636 . 2  |-  ( E. x  e.  ( Fil `  B ) A  C_  x  ->  -.  (/)  e.  ( fi `  A ) )
4434, 43impbid1 196 1  |-  ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  -> 
( -.  (/)  e.  ( fi `  A )  <->  E. x  e.  ( Fil `  B ) A 
C_  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519   _Vcvv 2763    C_ wss 3127   (/)c0 3430   ~Pcpw 3599   {csn 3614   ` cfv 4673  (class class class)co 5792   ficfi 7132   fBascfbas 17481   filGencfg 17482   Filcfil 17503
This theorem is referenced by:  cnfilca  24924
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-en 6832  df-fin 6835  df-fi 7133  df-fbas 17483  df-fg 17484  df-fil 17504
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