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Theorem ufilen 19478
Description: Any infinite set has an ultrafilter on it whose elements are of the same cardinality as the set. Any such ultrafilter is necessarily free. (Contributed by Jeff Hankins, 7-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.)
Assertion
Ref Expression
ufilen  |-  ( om  ~<_  X  ->  E. f  e.  ( UFil `  X
) A. x  e.  f  x  ~~  X
)
Distinct variable group:    x, f, X

Proof of Theorem ufilen
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 reldom 7308 . . . . . 6  |-  Rel  ~<_
21brrelex2i 4875 . . . . 5  |-  ( om  ~<_  X  ->  X  e.  _V )
3 numth3 8631 . . . . 5  |-  ( X  e.  _V  ->  X  e.  dom  card )
42, 3syl 16 . . . 4  |-  ( om  ~<_  X  ->  X  e.  dom  card )
5 csdfil 19442 . . . 4  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  { y  e.  ~P X  |  ( X  \  y )  ~<  X }  e.  ( Fil `  X
) )
64, 5mpancom 669 . . 3  |-  ( om  ~<_  X  ->  { y  e.  ~P X  |  ( X  \  y ) 
~<  X }  e.  ( Fil `  X ) )
7 filssufil 19460 . . 3  |-  ( { y  e.  ~P X  |  ( X  \ 
y )  ~<  X }  e.  ( Fil `  X
)  ->  E. f  e.  ( UFil `  X
) { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f
)
86, 7syl 16 . 2  |-  ( om  ~<_  X  ->  E. f  e.  ( UFil `  X
) { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f
)
9 elfvex 5712 . . . . . . 7  |-  ( f  e.  ( UFil `  X
)  ->  X  e.  _V )
109ad2antlr 726 . . . . . 6  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  X  e.  _V )
11 ufilfil 19452 . . . . . . . 8  |-  ( f  e.  ( UFil `  X
)  ->  f  e.  ( Fil `  X ) )
12 filelss 19400 . . . . . . . 8  |-  ( ( f  e.  ( Fil `  X )  /\  x  e.  f )  ->  x  C_  X )
1311, 12sylan 471 . . . . . . 7  |-  ( ( f  e.  ( UFil `  X )  /\  x  e.  f )  ->  x  C_  X )
1413adantll 713 . . . . . 6  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  x  C_  X )
15 ssdomg 7347 . . . . . 6  |-  ( X  e.  _V  ->  (
x  C_  X  ->  x  ~<_  X ) )
1610, 14, 15sylc 60 . . . . 5  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  x  ~<_  X )
17 filfbas 19396 . . . . . . . . 9  |-  ( f  e.  ( Fil `  X
)  ->  f  e.  ( fBas `  X )
)
1811, 17syl 16 . . . . . . . 8  |-  ( f  e.  ( UFil `  X
)  ->  f  e.  ( fBas `  X )
)
1918adantl 466 . . . . . . 7  |-  ( ( om  ~<_  X  /\  f  e.  ( UFil `  X
) )  ->  f  e.  ( fBas `  X
) )
20 fbncp 19387 . . . . . . 7  |-  ( ( f  e.  ( fBas `  X )  /\  x  e.  f )  ->  -.  ( X  \  x
)  e.  f )
2119, 20sylan 471 . . . . . 6  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  -.  ( X  \  x )  e.  f )
22 difss 3478 . . . . . . . . . . . . . 14  |-  ( X 
\  x )  C_  X
23 elpw2g 4450 . . . . . . . . . . . . . 14  |-  ( X  e.  _V  ->  (
( X  \  x
)  e.  ~P X  <->  ( X  \  x ) 
C_  X ) )
2422, 23mpbiri 233 . . . . . . . . . . . . 13  |-  ( X  e.  _V  ->  ( X  \  x )  e. 
~P X )
25243ad2ant1 1009 . . . . . . . . . . . 12  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  x )  e. 
~P X )
26 simp2 989 . . . . . . . . . . . . . 14  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  x  C_  X )
27 dfss4 3579 . . . . . . . . . . . . . 14  |-  ( x 
C_  X  <->  ( X  \  ( X  \  x
) )  =  x )
2826, 27sylib 196 . . . . . . . . . . . . 13  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  ( X  \  x ) )  =  x )
29 simp3 990 . . . . . . . . . . . . 13  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  x  ~<  X )
3028, 29eqbrtrd 4307 . . . . . . . . . . . 12  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  ( X  \  x ) )  ~<  X )
31 difeq2 3463 . . . . . . . . . . . . . 14  |-  ( y  =  ( X  \  x )  ->  ( X  \  y )  =  ( X  \  ( X  \  x ) ) )
3231breq1d 4297 . . . . . . . . . . . . 13  |-  ( y  =  ( X  \  x )  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  ( X  \  x
) )  ~<  X ) )
3332elrab 3112 . . . . . . . . . . . 12  |-  ( ( X  \  x )  e.  { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  <->  ( ( X  \  x )  e. 
~P X  /\  ( X  \  ( X  \  x ) )  ~<  X ) )
3425, 30, 33sylanbrc 664 . . . . . . . . . . 11  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  x )  e. 
{ y  e.  ~P X  |  ( X  \  y )  ~<  X }
)
35 ssel 3345 . . . . . . . . . . 11  |-  ( { y  e.  ~P X  |  ( X  \ 
y )  ~<  X }  C_  f  ->  ( ( X  \  x )  e. 
{ y  e.  ~P X  |  ( X  \  y )  ~<  X }  ->  ( X  \  x
)  e.  f ) )
3634, 35syl5com 30 . . . . . . . . . 10  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f  ->  ( X  \  x )  e.  f ) )
37363expa 1187 . . . . . . . . 9  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  x  ~<  X )  ->  ( { y  e.  ~P X  | 
( X  \  y
)  ~<  X }  C_  f  ->  ( X  \  x )  e.  f ) )
3837impancom 440 . . . . . . . 8  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f )  ->  (
x  ~<  X  ->  ( X  \  x )  e.  f ) )
3938con3d 133 . . . . . . 7  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f )  ->  ( -.  ( X  \  x
)  e.  f  ->  -.  x  ~<  X ) )
4039impancom 440 . . . . . 6  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  -.  ( X  \  x )  e.  f )  ->  ( {
y  e.  ~P X  |  ( X  \ 
y )  ~<  X }  C_  f  ->  -.  x  ~<  X ) )
4110, 14, 21, 40syl21anc 1217 . . . . 5  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  ( { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f  ->  -.  x  ~<  X ) )
42 bren2 7332 . . . . . 6  |-  ( x 
~~  X  <->  ( x  ~<_  X  /\  -.  x  ~<  X ) )
4342simplbi2 625 . . . . 5  |-  ( x  ~<_  X  ->  ( -.  x  ~<  X  ->  x  ~~  X ) )
4416, 41, 43sylsyld 56 . . . 4  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  ( { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f  ->  x  ~~  X ) )
4544ralrimdva 2801 . . 3  |-  ( ( om  ~<_  X  /\  f  e.  ( UFil `  X
) )  ->  ( { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f  ->  A. x  e.  f  x  ~~  X ) )
4645reximdva 2823 . 2  |-  ( om  ~<_  X  ->  ( E. f  e.  ( UFil `  X ) { y  e.  ~P X  | 
( X  \  y
)  ~<  X }  C_  f  ->  E. f  e.  (
UFil `  X ) A. x  e.  f  x  ~~  X ) )
478, 46mpd 15 1  |-  ( om  ~<_  X  ->  E. f  e.  ( UFil `  X
) A. x  e.  f  x  ~~  X
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   E.wrex 2711   {crab 2714   _Vcvv 2967    \ cdif 3320    C_ wss 3323   ~Pcpw 3855   class class class wbr 4287   dom cdm 4835   ` cfv 5413   omcom 6471    ~~ cen 7299    ~<_ cdom 7300    ~< csdm 7301   cardccrd 8097   fBascfbas 17779   Filcfil 19393   UFilcufil 19447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-ac2 8624
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-rpss 6355  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fi 7653  df-oi 7716  df-card 8101  df-ac 8278  df-cda 8329  df-fbas 17789  df-fg 17790  df-fil 19394  df-ufil 19449
This theorem is referenced by: (None)
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