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Theorem ufilen 17625
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 6869 . . . . . 6  |-  Rel  ~<_
21brrelex2i 4730 . . . . 5  |-  ( om  ~<_  X  ->  X  e.  _V )
3 numth3 8097 . . . . 5  |-  ( X  e.  _V  ->  X  e.  dom  card )
42, 3syl 15 . . . 4  |-  ( om  ~<_  X  ->  X  e.  dom  card )
5 csdfil 17589 . . . 4  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  { y  e.  ~P X  |  ( X  \  y )  ~<  X }  e.  ( Fil `  X
) )
64, 5mpancom 650 . . 3  |-  ( om  ~<_  X  ->  { y  e.  ~P X  |  ( X  \  y ) 
~<  X }  e.  ( Fil `  X ) )
7 filssufil 17607 . . 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 15 . 2  |-  ( om  ~<_  X  ->  E. f  e.  ( UFil `  X
) { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f
)
9 elfvex 5555 . . . . . . 7  |-  ( f  e.  ( UFil `  X
)  ->  X  e.  _V )
109ad2antlr 707 . . . . . 6  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  X  e.  _V )
11 ufilfil 17599 . . . . . . . 8  |-  ( f  e.  ( UFil `  X
)  ->  f  e.  ( Fil `  X ) )
12 filelss 17547 . . . . . . . 8  |-  ( ( f  e.  ( Fil `  X )  /\  x  e.  f )  ->  x  C_  X )
1311, 12sylan 457 . . . . . . 7  |-  ( ( f  e.  ( UFil `  X )  /\  x  e.  f )  ->  x  C_  X )
1413adantll 694 . . . . . 6  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  x  C_  X )
15 ssdomg 6907 . . . . . 6  |-  ( X  e.  _V  ->  (
x  C_  X  ->  x  ~<_  X ) )
1610, 14, 15sylc 56 . . . . 5  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  x  ~<_  X )
17 filfbas 17543 . . . . . . . . 9  |-  ( f  e.  ( Fil `  X
)  ->  f  e.  ( fBas `  X )
)
1811, 17syl 15 . . . . . . . 8  |-  ( f  e.  ( UFil `  X
)  ->  f  e.  ( fBas `  X )
)
1918adantl 452 . . . . . . 7  |-  ( ( om  ~<_  X  /\  f  e.  ( UFil `  X
) )  ->  f  e.  ( fBas `  X
) )
20 fbncp 17534 . . . . . . 7  |-  ( ( f  e.  ( fBas `  X )  /\  x  e.  f )  ->  -.  ( X  \  x
)  e.  f )
2119, 20sylan 457 . . . . . 6  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  -.  ( X  \  x )  e.  f )
22 difss 3303 . . . . . . . . . . . . . 14  |-  ( X 
\  x )  C_  X
23 elpw2g 4174 . . . . . . . . . . . . . 14  |-  ( X  e.  _V  ->  (
( X  \  x
)  e.  ~P X  <->  ( X  \  x ) 
C_  X ) )
2422, 23mpbiri 224 . . . . . . . . . . . . 13  |-  ( X  e.  _V  ->  ( X  \  x )  e. 
~P X )
25243ad2ant1 976 . . . . . . . . . . . 12  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  x )  e. 
~P X )
26 simp2 956 . . . . . . . . . . . . . 14  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  x  C_  X )
27 dfss4 3403 . . . . . . . . . . . . . 14  |-  ( x 
C_  X  <->  ( X  \  ( X  \  x
) )  =  x )
2826, 27sylib 188 . . . . . . . . . . . . 13  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  ( X  \  x ) )  =  x )
29 simp3 957 . . . . . . . . . . . . 13  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  x  ~<  X )
3028, 29eqbrtrd 4043 . . . . . . . . . . . 12  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  ( X  \  x ) )  ~<  X )
31 difeq2 3288 . . . . . . . . . . . . . 14  |-  ( y  =  ( X  \  x )  ->  ( X  \  y )  =  ( X  \  ( X  \  x ) ) )
3231breq1d 4033 . . . . . . . . . . . . 13  |-  ( y  =  ( X  \  x )  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  ( X  \  x
) )  ~<  X ) )
3332elrab 2923 . . . . . . . . . . . 12  |-  ( ( X  \  x )  e.  { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  <->  ( ( X  \  x )  e. 
~P X  /\  ( X  \  ( X  \  x ) )  ~<  X ) )
3425, 30, 33sylanbrc 645 . . . . . . . . . . 11  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  x )  e. 
{ y  e.  ~P X  |  ( X  \  y )  ~<  X }
)
35 ssel 3174 . . . . . . . . . . 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 26 . . . . . . . . . 10  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f  ->  ( X  \  x )  e.  f ) )
37363expa 1151 . . . . . . . . 9  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  x  ~<  X )  ->  ( { y  e.  ~P X  | 
( X  \  y
)  ~<  X }  C_  f  ->  ( X  \  x )  e.  f ) )
3837impancom 427 . . . . . . . 8  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f )  ->  (
x  ~<  X  ->  ( X  \  x )  e.  f ) )
3938con3d 125 . . . . . . 7  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f )  ->  ( -.  ( X  \  x
)  e.  f  ->  -.  x  ~<  X ) )
4039impancom 427 . . . . . 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 1181 . . . . 5  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  ( { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f  ->  -.  x  ~<  X ) )
42 bren2 6892 . . . . . 6  |-  ( x 
~~  X  <->  ( x  ~<_  X  /\  -.  x  ~<  X ) )
4342simplbi2 608 . . . . 5  |-  ( x  ~<_  X  ->  ( -.  x  ~<  X  ->  x  ~~  X ) )
4416, 41, 43sylsyld 52 . . . 4  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  ( { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f  ->  x  ~~  X ) )
4544ralrimdva 2633 . . 3  |-  ( ( om  ~<_  X  /\  f  e.  ( UFil `  X
) )  ->  ( { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f  ->  A. x  e.  f  x  ~~  X ) )
4645reximdva 2655 . 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 14 1  |-  ( om  ~<_  X  ->  E. f  e.  ( UFil `  X
) A. x  e.  f  x  ~~  X
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023   omcom 4656   dom cdm 4689   ` cfv 5255    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568   fBascfbas 17518   Filcfil 17540   UFilcufil 17594
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-ac2 8089
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-rpss 6277  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-oi 7225  df-card 7572  df-ac 7743  df-cda 7794  df-fbas 17520  df-fg 17521  df-fil 17541  df-ufil 17596
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