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Theorem efilcp 24963
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 17555 . . . . . . . 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 3484 . . . . . . 7  |-  (/)  C_  { B }
4 sseq2 3201 . . . . . . . 8  |-  ( x  =  { B }  ->  ( (/)  C_  x  <->  (/)  C_  { B } ) )
54rspcev 2885 . . . . . . 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 3200 . . . . . . 7  |-  ( A  =  (/)  ->  ( A 
C_  x  <->  (/)  C_  x
) )
87rexbidv 2565 . . . . . 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 17558 . . . . . . . 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 17569 . . . . . 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 4193 . . . . . . . 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 4161 . . . . . . . 8  |-  ( ( A  C_  ~P B  /\  ~P B  e.  _V )  ->  A  e.  _V )
24 ssfii 7168 . . . . . . . 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 17563 . . . . . . 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 3190 . . . . 5  |-  ( ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  /\  ( A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  C_  ( B filGen ( fi `  A ) ) )
30 sseq2 3201 . . . . . 6  |-  ( x  =  ( B filGen ( fi `  A ) )  ->  ( A  C_  x  <->  A  C_  ( B
filGen ( fi `  A
) ) ) )
3130rspcev 2885 . . . . 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 2525 . 2  |-  ( ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  -> 
( -.  (/)  e.  ( fi `  A )  ->  E. x  e.  ( Fil `  B ) A  C_  x )
)
35 filfbas 17539 . . . 4  |-  ( x  e.  ( Fil `  B
)  ->  x  e.  ( fBas `  B )
)
36 fbasfip 17559 . . . 4  |-  ( x  e.  ( fBas `  B
)  ->  -.  (/)  e.  ( fi `  x ) )
37 vex 2792 . . . . . . 7  |-  x  e. 
_V
38 fiss 7173 . . . . . . 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 3180 . . . . 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 2662 . 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 1623    e. wcel 1685    =/= wne 2447   E.wrex 2545   _Vcvv 2789    C_ wss 3153   (/)c0 3456   ~Pcpw 3626   {csn 3641   ` cfv 5221  (class class class)co 5820   ficfi 7160   fBascfbas 17514   filGencfg 17515   Filcfil 17536
This theorem is referenced by:  cnfilca  24967
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-en 6860  df-fin 6863  df-fi 7161  df-fbas 17516  df-fg 17517  df-fil 17537
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