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Theorem efilcp 25563
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 17561 . . . . . . . 8  |-  ( ( B  e.  V  /\  B  =/=  (/) )  ->  { B }  e.  ( Fil `  B ) )
213adant1 973 . . . . . . 7  |-  ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  ->  { B }  e.  ( Fil `  B ) )
3 0ss 3485 . . . . . . 7  |-  (/)  C_  { B }
4 sseq2 3202 . . . . . . . 8  |-  ( x  =  { B }  ->  ( (/)  C_  x  <->  (/)  C_  { B } ) )
54rspcev 2886 . . . . . . 7  |-  ( ( { B }  e.  ( Fil `  B )  /\  (/)  C_  { B } )  ->  E. x  e.  ( Fil `  B
) (/)  C_  x )
62, 3, 5sylancl 643 . . . . . 6  |-  ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  ->  E. x  e.  ( Fil `  B ) (/)  C_  x )
7 sseq1 3201 . . . . . . 7  |-  ( A  =  (/)  ->  ( A 
C_  x  <->  (/)  C_  x
) )
87rexbidv 2566 . . . . . 6  |-  ( A  =  (/)  ->  ( E. x  e.  ( Fil `  B ) A  C_  x 
<->  E. x  e.  ( Fil `  B )
(/)  C_  x ) )
96, 8syl5ibrcom 213 . . . . 5  |-  ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  -> 
( A  =  (/)  ->  E. x  e.  ( Fil `  B ) A  C_  x )
)
109imp 418 . . . 4  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  A  =  (/) )  ->  E. x  e.  ( Fil `  B ) A 
C_  x )
1110a1d 22 . . 3  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  A  =  (/) )  -> 
( -.  (/)  e.  ( fi `  A )  ->  E. x  e.  ( Fil `  B ) A  C_  x )
)
12 simpl1 958 . . . . . . 7  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  C_  ~P B )
13 simprl 732 . . . . . . 7  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  =/=  (/) )
14 simprr 733 . . . . . . 7  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  -.  (/)  e.  ( fi `  A ) )
15 simpl2 959 . . . . . . . 8  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  B  e.  V
)
16 fsubbas 17564 . . . . . . . 8  |-  ( B  e.  V  ->  (
( fi `  A
)  e.  ( fBas `  B )  <->  ( A  C_ 
~P B  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi `  A ) ) ) )
1715, 16syl 15 . . . . . . 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 1135 . . . . . 6  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( fi `  A )  e.  (
fBas `  B )
)
19 fgcl 17575 . . . . . 6  |-  ( ( fi `  A )  e.  ( fBas `  B
)  ->  ( B filGen ( fi `  A
) )  e.  ( Fil `  B ) )
2018, 19syl 15 . . . . 5  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( B filGen ( fi `  A ) )  e.  ( Fil `  B ) )
21 pwexg 4196 . . . . . . . 8  |-  ( B  e.  V  ->  ~P B  e.  _V )
2215, 21syl 15 . . . . . . 7  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ~P B  e. 
_V )
23 ssexg 4162 . . . . . . . 8  |-  ( ( A  C_  ~P B  /\  ~P B  e.  _V )  ->  A  e.  _V )
24 ssfii 7174 . . . . . . . 8  |-  ( A  e.  _V  ->  A  C_  ( fi `  A
) )
2523, 24syl 15 . . . . . . 7  |-  ( ( A  C_  ~P B  /\  ~P B  e.  _V )  ->  A  C_  ( fi `  A ) )
2612, 22, 25syl2anc 642 . . . . . 6  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  C_  ( fi `  A ) )
27 ssfg 17569 . . . . . . 7  |-  ( ( fi `  A )  e.  ( fBas `  B
)  ->  ( fi `  A )  C_  ( B filGen ( fi `  A ) ) )
2818, 27syl 15 . . . . . 6  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( fi `  A )  C_  ( B filGen ( fi `  A ) ) )
2926, 28sstrd 3191 . . . . 5  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  C_  ( B filGen ( fi `  A ) ) )
30 sseq2 3202 . . . . . 6  |-  ( x  =  ( B filGen ( fi `  A ) )  ->  ( A  C_  x  <->  A  C_  ( B
filGen ( fi `  A
) ) ) )
3130rspcev 2886 . . . . 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 642 . . . 4  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  E. x  e.  ( Fil `  B ) A  C_  x )
3332expr 598 . . 3  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  A  =/=  (/) )  ->  ( -.  (/)  e.  ( fi
`  A )  ->  E. x  e.  ( Fil `  B ) A 
C_  x ) )
3411, 33pm2.61dane 2526 . 2  |-  ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  -> 
( -.  (/)  e.  ( fi `  A )  ->  E. x  e.  ( Fil `  B ) A  C_  x )
)
35 filfbas 17545 . . . 4  |-  ( x  e.  ( Fil `  B
)  ->  x  e.  ( fBas `  B )
)
36 fbasfip 17565 . . . 4  |-  ( x  e.  ( fBas `  B
)  ->  -.  (/)  e.  ( fi `  x ) )
37 vex 2793 . . . . . . 7  |-  x  e. 
_V
38 fiss 7179 . . . . . . 7  |-  ( ( x  e.  _V  /\  A  C_  x )  -> 
( fi `  A
)  C_  ( fi `  x ) )
3937, 38mpan 651 . . . . . 6  |-  ( A 
C_  x  ->  ( fi `  A )  C_  ( fi `  x ) )
4039sseld 3181 . . . . 5  |-  ( A 
C_  x  ->  ( (/) 
e.  ( fi `  A )  ->  (/)  e.  ( fi `  x ) ) )
4140con3rr3 128 . . . 4  |-  ( -.  (/)  e.  ( fi `  x )  ->  ( A  C_  x  ->  -.  (/) 
e.  ( fi `  A ) ) )
4235, 36, 413syl 18 . . 3  |-  ( x  e.  ( Fil `  B
)  ->  ( A  C_  x  ->  -.  (/)  e.  ( fi `  A ) ) )
4342rexlimiv 2663 . 2  |-  ( E. x  e.  ( Fil `  B ) A  C_  x  ->  -.  (/)  e.  ( fi `  A ) )
4434, 43impbid1 194 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 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   E.wrex 2546   _Vcvv 2790    C_ wss 3154   (/)c0 3457   ~Pcpw 3627   {csn 3642   ` cfv 5257  (class class class)co 5860   ficfi 7166   fBascfbas 17520   filGencfg 17521   Filcfil 17542
This theorem is referenced by:  cnfilca  25567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-en 6866  df-fin 6869  df-fi 7167  df-fbas 17522  df-fg 17523  df-fil 17543
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