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Theorem uffixfr 17947
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 444 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  e.  ( UFil `  X ) )
2 ufilfil 17928 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
3 filtop 17879 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
42, 3syl 16 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  X  e.  F )
54adantr 452 . . . . . 6  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  X  e.  F )
6 filn0 17886 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
7 intssuni 4064 . . . . . . . . 9  |-  ( F  =/=  (/)  ->  |^| F  C_  U. F )
82, 6, 73syl 19 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  U. F )
9 filunibas 17905 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
102, 9syl 16 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  U. F  =  X )
118, 10sseqtrd 3376 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  X )
1211sselda 3340 . . . . . 6  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  X )
13 uffix 17945 . . . . . 6  |-  ( ( X  e.  F  /\  A  e.  X )  ->  ( { { A } }  e.  ( fBas `  X )  /\  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } } ) ) )
145, 12, 13syl2anc 643 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  -> 
( { { A } }  e.  ( fBas `  X )  /\  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } } ) ) )
1514simprd 450 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } } ) )
1614simpld 446 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { { A } }  e.  ( fBas `  X
) )
17 fgcl 17902 . . . . 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 2509 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { x  e.  ~P X  |  A  e.  x }  e.  ( Fil `  X ) )
202adantr 452 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  e.  ( Fil `  X ) )
21 filsspw 17875 . . . . 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 4050 . . . . . 6  |-  ( A  e.  |^| F  ->  ( A  e.  |^| F  <->  A. x  e.  F  A  e.  x ) )
2423ibi 233 . . . . 5  |-  ( A  e.  |^| F  ->  A. x  e.  F  A  e.  x )
2524adantl 453 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A. x  e.  F  A  e.  x )
26 ssrab 3413 . . . 4  |-  ( F 
C_  { x  e. 
~P X  |  A  e.  x }  <->  ( F  C_ 
~P X  /\  A. x  e.  F  A  e.  x ) )
2722, 25, 26sylanbrc 646 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  C_  { x  e. 
~P X  |  A  e.  x } )
28 ufilmax 17931 . . 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 1184 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  =  { x  e.  ~P X  |  A  e.  x } )
30 eqimss 3392 . . . . 5  |-  ( F  =  { x  e. 
~P X  |  A  e.  x }  ->  F  C_ 
{ x  e.  ~P X  |  A  e.  x } )
3130adantl 453 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  F  C_ 
{ x  e.  ~P X  |  A  e.  x } )
3226simprbi 451 . . . 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 2496 . . . . . 6  |-  ( F  =  { x  e. 
~P X  |  A  e.  x }  ->  ( X  e.  F  <->  X  e.  { x  e.  ~P X  |  A  e.  x } ) )
3534biimpac 473 . . . . 5  |-  ( ( X  e.  F  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  X  e.  { x  e.  ~P X  |  A  e.  x } )
364, 35sylan 458 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  X  e.  { x  e.  ~P X  |  A  e.  x } )
37 eleq2 2496 . . . . . 6  |-  ( x  =  X  ->  ( A  e.  x  <->  A  e.  X ) )
3837elrab 3084 . . . . 5  |-  ( X  e.  { x  e. 
~P X  |  A  e.  x }  <->  ( X  e.  ~P X  /\  A  e.  X ) )
3938simprbi 451 . . . 4  |-  ( X  e.  { x  e. 
~P X  |  A  e.  x }  ->  A  e.  X )
40 elintg 4050 . . . 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 224 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  A  e.  |^| F )
4329, 42impbida 806 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   {crab 2701    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   {csn 3806   U.cuni 4007   |^|cint 4042   ` cfv 5446  (class class class)co 6073   fBascfbas 16681   filGencfg 16682   Filcfil 17869   UFilcufil 17923
This theorem is referenced by:  uffix2  17948  uffixsn  17949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-fbas 16691  df-fg 16692  df-fil 17870  df-ufil 17925
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