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Theorem uffixsn 17620
Description: The singleton of the generator of a fixed ultrafilter is in the filter. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
uffixsn  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  F
)

Proof of Theorem uffixsn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ufilfil 17599 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
2 filn0 17557 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
3 intssuni 3884 . . . . . . . 8  |-  ( F  =/=  (/)  ->  |^| F  C_  U. F )
41, 2, 33syl 18 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  U. F )
5 filunibas 17576 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
61, 5syl 15 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  U. F  =  X )
74, 6sseqtrd 3214 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  X )
87sselda 3180 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  X )
98snssd 3760 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  C_  X
)
10 snex 4216 . . . . 5  |-  { A }  e.  _V
1110elpw 3631 . . . 4  |-  ( { A }  e.  ~P X 
<->  { A }  C_  X )
129, 11sylibr 203 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  ~P X )
13 snidg 3665 . . . 4  |-  ( A  e.  |^| F  ->  A  e.  { A } )
1413adantl 452 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  { A } )
15 eleq2 2344 . . . 4  |-  ( x  =  { A }  ->  ( A  e.  x  <->  A  e.  { A }
) )
1615elrab 2923 . . 3  |-  ( { A }  e.  {
x  e.  ~P X  |  A  e.  x } 
<->  ( { A }  e.  ~P X  /\  A  e.  { A } ) )
1712, 14, 16sylanbrc 645 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  {
x  e.  ~P X  |  A  e.  x } )
18 uffixfr 17618 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( A  e.  |^| F  <->  F  =  { x  e.  ~P X  |  A  e.  x } ) )
1918biimpa 470 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  =  { x  e.  ~P X  |  A  e.  x } )
2017, 19eleqtrrd 2360 1  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   U.cuni 3827   |^|cint 3862   ` cfv 5255   Filcfil 17540   UFilcufil 17594
This theorem is referenced by:  ufildom1  17621  cfinufil  17623  fin1aufil  17627
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-fbas 17520  df-fg 17521  df-fil 17541  df-ufil 17596
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