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Theorem uffixfr 17960
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 445 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  e.  ( UFil `  X ) )
2 ufilfil 17941 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
3 filtop 17892 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
42, 3syl 16 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  X  e.  F )
54adantr 453 . . . . . 6  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  X  e.  F )
6 filn0 17899 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
7 intssuni 4074 . . . . . . . . 9  |-  ( F  =/=  (/)  ->  |^| F  C_  U. F )
82, 6, 73syl 19 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  U. F )
9 filunibas 17918 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
102, 9syl 16 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  U. F  =  X )
118, 10sseqtrd 3386 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  X )
1211sselda 3350 . . . . . 6  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  X )
13 uffix 17958 . . . . . 6  |-  ( ( X  e.  F  /\  A  e.  X )  ->  ( { { A } }  e.  ( fBas `  X )  /\  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } } ) ) )
145, 12, 13syl2anc 644 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  -> 
( { { A } }  e.  ( fBas `  X )  /\  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } } ) ) )
1514simprd 451 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } } ) )
1614simpld 447 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { { A } }  e.  ( fBas `  X
) )
17 fgcl 17915 . . . . 5  |-  ( { { A } }  e.  ( fBas `  X
)  ->  ( X filGen { { A } } )  e.  ( Fil `  X ) )
1816, 17syl 16 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  -> 
( X filGen { { A } } )  e.  ( Fil `  X
) )
1915, 18eqeltrd 2512 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { x  e.  ~P X  |  A  e.  x }  e.  ( Fil `  X ) )
202adantr 453 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  e.  ( Fil `  X ) )
21 filsspw 17888 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  C_  ~P X )
2220, 21syl 16 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  C_  ~P X )
23 elintg 4060 . . . . . 6  |-  ( A  e.  |^| F  ->  ( A  e.  |^| F  <->  A. x  e.  F  A  e.  x ) )
2423ibi 234 . . . . 5  |-  ( A  e.  |^| F  ->  A. x  e.  F  A  e.  x )
2524adantl 454 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A. x  e.  F  A  e.  x )
26 ssrab 3423 . . . 4  |-  ( F 
C_  { x  e. 
~P X  |  A  e.  x }  <->  ( F  C_ 
~P X  /\  A. x  e.  F  A  e.  x ) )
2722, 25, 26sylanbrc 647 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  C_  { x  e. 
~P X  |  A  e.  x } )
28 ufilmax 17944 . . 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 1185 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  =  { x  e.  ~P X  |  A  e.  x } )
30 eqimss 3402 . . . . 5  |-  ( F  =  { x  e. 
~P X  |  A  e.  x }  ->  F  C_ 
{ x  e.  ~P X  |  A  e.  x } )
3130adantl 454 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  F  C_ 
{ x  e.  ~P X  |  A  e.  x } )
3226simprbi 452 . . . 4  |-  ( F 
C_  { x  e. 
~P X  |  A  e.  x }  ->  A. x  e.  F  A  e.  x )
3331, 32syl 16 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  A. x  e.  F  A  e.  x )
34 eleq2 2499 . . . . . 6  |-  ( F  =  { x  e. 
~P X  |  A  e.  x }  ->  ( X  e.  F  <->  X  e.  { x  e.  ~P X  |  A  e.  x } ) )
3534biimpac 474 . . . . 5  |-  ( ( X  e.  F  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  X  e.  { x  e.  ~P X  |  A  e.  x } )
364, 35sylan 459 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  X  e.  { x  e.  ~P X  |  A  e.  x } )
37 eleq2 2499 . . . . . 6  |-  ( x  =  X  ->  ( A  e.  x  <->  A  e.  X ) )
3837elrab 3094 . . . . 5  |-  ( X  e.  { x  e. 
~P X  |  A  e.  x }  <->  ( X  e.  ~P X  /\  A  e.  X ) )
3938simprbi 452 . . . 4  |-  ( X  e.  { x  e. 
~P X  |  A  e.  x }  ->  A  e.  X )
40 elintg 4060 . . . 4  |-  ( A  e.  X  ->  ( A  e.  |^| F  <->  A. x  e.  F  A  e.  x ) )
4136, 39, 403syl 19 . . 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 225 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  A  e.  |^| F )
4329, 42impbida 807 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   U.cuni 4017   |^|cint 4052   ` cfv 5457  (class class class)co 6084   fBascfbas 16694   filGencfg 16695   Filcfil 17882   UFilcufil 17936
This theorem is referenced by:  uffix2  17961  uffixsn  17962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-fbas 16704  df-fg 16705  df-fil 17883  df-ufil 17938
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