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Theorem uffixfr 17634
Description: An ultrafilter is either fixed or free. A fixed ultrafilter is called principal (generated by a single element  A), and a free ultrafilter is called nonprincipal (having empty intersection). Note that examples of free ultrafilters cannot be defined in ZFC without some form of global choice. (Contributed by Jeff Hankins, 4-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
uffixfr  |-  ( F  e.  ( UFil `  X
)  ->  ( A  e.  |^| F  <->  F  =  { x  e.  ~P X  |  A  e.  x } ) )
Distinct variable groups:    x, A    x, F    x, X

Proof of Theorem uffixfr
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  e.  ( UFil `  X ) )
2 ufilfil 17615 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
3 filtop 17566 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
42, 3syl 15 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  X  e.  F )
54adantr 451 . . . . . 6  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  X  e.  F )
6 filn0 17573 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
7 intssuni 3900 . . . . . . . . 9  |-  ( F  =/=  (/)  ->  |^| F  C_  U. F )
82, 6, 73syl 18 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  U. F )
9 filunibas 17592 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
102, 9syl 15 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  U. F  =  X )
118, 10sseqtrd 3227 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  X )
1211sselda 3193 . . . . . 6  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  X )
13 uffix 17632 . . . . . 6  |-  ( ( X  e.  F  /\  A  e.  X )  ->  ( { { A } }  e.  ( fBas `  X )  /\  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } } ) ) )
145, 12, 13syl2anc 642 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  -> 
( { { A } }  e.  ( fBas `  X )  /\  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } } ) ) )
1514simprd 449 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } } ) )
1614simpld 445 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { { A } }  e.  ( fBas `  X
) )
17 fgcl 17589 . . . . 5  |-  ( { { A } }  e.  ( fBas `  X
)  ->  ( X filGen { { A } } )  e.  ( Fil `  X ) )
1816, 17syl 15 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  -> 
( X filGen { { A } } )  e.  ( Fil `  X
) )
1915, 18eqeltrd 2370 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { x  e.  ~P X  |  A  e.  x }  e.  ( Fil `  X ) )
202adantr 451 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  e.  ( Fil `  X ) )
21 filsspw 17562 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  C_  ~P X )
2220, 21syl 15 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  C_  ~P X )
23 elintg 3886 . . . . . 6  |-  ( A  e.  |^| F  ->  ( A  e.  |^| F  <->  A. x  e.  F  A  e.  x ) )
2423ibi 232 . . . . 5  |-  ( A  e.  |^| F  ->  A. x  e.  F  A  e.  x )
2524adantl 452 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A. x  e.  F  A  e.  x )
26 ssrab 3264 . . . 4  |-  ( F 
C_  { x  e. 
~P X  |  A  e.  x }  <->  ( F  C_ 
~P X  /\  A. x  e.  F  A  e.  x ) )
2722, 25, 26sylanbrc 645 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  C_  { x  e. 
~P X  |  A  e.  x } )
28 ufilmax 17618 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  {
x  e.  ~P X  |  A  e.  x }  e.  ( Fil `  X )  /\  F  C_ 
{ x  e.  ~P X  |  A  e.  x } )  ->  F  =  { x  e.  ~P X  |  A  e.  x } )
291, 19, 27, 28syl3anc 1182 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  =  { x  e.  ~P X  |  A  e.  x } )
30 eqimss 3243 . . . . 5  |-  ( F  =  { x  e. 
~P X  |  A  e.  x }  ->  F  C_ 
{ x  e.  ~P X  |  A  e.  x } )
3130adantl 452 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  F  C_ 
{ x  e.  ~P X  |  A  e.  x } )
3226simprbi 450 . . . 4  |-  ( F 
C_  { x  e. 
~P X  |  A  e.  x }  ->  A. x  e.  F  A  e.  x )
3331, 32syl 15 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  A. x  e.  F  A  e.  x )
34 eleq2 2357 . . . . . 6  |-  ( F  =  { x  e. 
~P X  |  A  e.  x }  ->  ( X  e.  F  <->  X  e.  { x  e.  ~P X  |  A  e.  x } ) )
3534biimpac 472 . . . . 5  |-  ( ( X  e.  F  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  X  e.  { x  e.  ~P X  |  A  e.  x } )
364, 35sylan 457 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  X  e.  { x  e.  ~P X  |  A  e.  x } )
37 eleq2 2357 . . . . . 6  |-  ( x  =  X  ->  ( A  e.  x  <->  A  e.  X ) )
3837elrab 2936 . . . . 5  |-  ( X  e.  { x  e. 
~P X  |  A  e.  x }  <->  ( X  e.  ~P X  /\  A  e.  X ) )
3938simprbi 450 . . . 4  |-  ( X  e.  { x  e. 
~P X  |  A  e.  x }  ->  A  e.  X )
40 elintg 3886 . . . 4  |-  ( A  e.  X  ->  ( A  e.  |^| F  <->  A. x  e.  F  A  e.  x ) )
4136, 39, 403syl 18 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  ( A  e.  |^| F  <->  A. x  e.  F  A  e.  x ) )
4233, 41mpbird 223 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  A  e.  |^| F )
4329, 42impbida 805 1  |-  ( F  e.  ( UFil `  X
)  ->  ( A  e.  |^| F  <->  F  =  { x  e.  ~P X  |  A  e.  x } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   U.cuni 3843   |^|cint 3878   ` cfv 5271  (class class class)co 5874   fBascfbas 17534   filGencfg 17535   Filcfil 17556   UFilcufil 17610
This theorem is referenced by:  uffix2  17635  uffixsn  17636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-fbas 17536  df-fg 17537  df-fil 17557  df-ufil 17612
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