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Theorem uffixsn 17833
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 17812 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
2 filn0 17770 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
3 intssuni 3986 . . . . . . . 8  |-  ( F  =/=  (/)  ->  |^| F  C_  U. F )
41, 2, 33syl 18 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  U. F )
5 filunibas 17789 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
61, 5syl 15 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  U. F  =  X )
74, 6sseqtrd 3300 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  X )
87sselda 3266 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  X )
98snssd 3858 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  C_  X
)
10 snex 4318 . . . . 5  |-  { A }  e.  _V
1110elpw 3720 . . . 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 3754 . . . 4  |-  ( A  e.  |^| F  ->  A  e.  { A } )
1413adantl 452 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  { A } )
15 eleq2 2427 . . . 4  |-  ( x  =  { A }  ->  ( A  e.  x  <->  A  e.  { A }
) )
1615elrab 3009 . . 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 17831 . . 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 2443 1  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   {crab 2632    C_ wss 3238   (/)c0 3543   ~Pcpw 3714   {csn 3729   U.cuni 3929   |^|cint 3964   ` cfv 5358   Filcfil 17753   UFilcufil 17807
This theorem is referenced by:  ufildom1  17834  cfinufil  17836  fin1aufil  17840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-int 3965  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-fbas 16590  df-fg 16591  df-fil 17754  df-ufil 17809
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