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Theorem uffixsn 17910
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 17889 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
2 filn0 17847 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
3 intssuni 4032 . . . . . . . 8  |-  ( F  =/=  (/)  ->  |^| F  C_  U. F )
41, 2, 33syl 19 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  U. F )
5 filunibas 17866 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
61, 5syl 16 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  U. F  =  X )
74, 6sseqtrd 3344 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  X )
87sselda 3308 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  X )
98snssd 3903 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  C_  X
)
10 snex 4365 . . . . 5  |-  { A }  e.  _V
1110elpw 3765 . . . 4  |-  ( { A }  e.  ~P X 
<->  { A }  C_  X )
129, 11sylibr 204 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  ~P X )
13 snidg 3799 . . . 4  |-  ( A  e.  |^| F  ->  A  e.  { A } )
1413adantl 453 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  { A } )
15 eleq2 2465 . . . 4  |-  ( x  =  { A }  ->  ( A  e.  x  <->  A  e.  { A }
) )
1615elrab 3052 . . 3  |-  ( { A }  e.  {
x  e.  ~P X  |  A  e.  x } 
<->  ( { A }  e.  ~P X  /\  A  e.  { A } ) )
1712, 14, 16sylanbrc 646 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  {
x  e.  ~P X  |  A  e.  x } )
18 uffixfr 17908 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( A  e.  |^| F  <->  F  =  { x  e.  ~P X  |  A  e.  x } ) )
1918biimpa 471 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  =  { x  e.  ~P X  |  A  e.  x } )
2017, 19eleqtrrd 2481 1  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   U.cuni 3975   |^|cint 4010   ` cfv 5413   Filcfil 17830   UFilcufil 17884
This theorem is referenced by:  ufildom1  17911  cfinufil  17913  fin1aufil  17917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-fbas 16654  df-fg 16655  df-fil 17831  df-ufil 17886
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