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Theorem uffixsn 20718
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 20697 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
2 filn0 20655 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
3 intssuni 4250 . . . . . . . 8  |-  ( F  =/=  (/)  ->  |^| F  C_  U. F )
41, 2, 33syl 18 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  U. F )
5 filunibas 20674 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
61, 5syl 17 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  U. F  =  X )
74, 6sseqtrd 3478 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  X )
87sselda 3442 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  X )
98snssd 4117 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  C_  X
)
10 snex 4632 . . . . 5  |-  { A }  e.  _V
1110elpw 3961 . . . 4  |-  ( { A }  e.  ~P X 
<->  { A }  C_  X )
129, 11sylibr 212 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  ~P X )
13 snidg 3998 . . . 4  |-  ( A  e.  |^| F  ->  A  e.  { A } )
1413adantl 464 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  { A } )
15 eleq2 2475 . . . 4  |-  ( x  =  { A }  ->  ( A  e.  x  <->  A  e.  { A }
) )
1615elrab 3207 . . 3  |-  ( { A }  e.  {
x  e.  ~P X  |  A  e.  x } 
<->  ( { A }  e.  ~P X  /\  A  e.  { A } ) )
1712, 14, 16sylanbrc 662 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  {
x  e.  ~P X  |  A  e.  x } )
18 uffixfr 20716 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( A  e.  |^| F  <->  F  =  { x  e.  ~P X  |  A  e.  x } ) )
1918biimpa 482 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  =  { x  e.  ~P X  |  A  e.  x } )
2017, 19eleqtrrd 2493 1  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  F
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   {crab 2758    C_ wss 3414   (/)c0 3738   ~Pcpw 3955   {csn 3972   U.cuni 4191   |^|cint 4227   ` cfv 5569   Filcfil 20638   UFilcufil 20692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-int 4228  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-fbas 18736  df-fg 18737  df-fil 20639  df-ufil 20694
This theorem is referenced by:  ufildom1  20719  cfinufil  20721  fin1aufil  20725
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