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Theorem ufildom1 17621
Description: An ultrafilter is generated by at most one element (because free ultrafilters have no generators and fixed ultrafilters have exactly one). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
ufildom1  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  ~<_  1o )

Proof of Theorem ufildom1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 breq1 4026 . 2  |-  ( |^| F  =  (/)  ->  ( |^| F  ~<_  1o  <->  (/)  ~<_  1o ) )
2 uffixsn 17620 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  { x }  e.  F )
3 intss1 3877 . . . . . . . . 9  |-  ( { x }  e.  F  ->  |^| F  C_  { x } )
42, 3syl 15 . . . . . . . 8  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  |^| F  C_  { x } )
5 simpr 447 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  x  e.  |^| F )
65snssd 3760 . . . . . . . 8  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  { x }  C_  |^| F )
74, 6eqssd 3196 . . . . . . 7  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  |^| F  =  { x } )
87ex 423 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  |^| F  ->  |^| F  =  { x } ) )
98eximdv 1608 . . . . 5  |-  ( F  e.  ( UFil `  X
)  ->  ( E. x  x  e.  |^| F  ->  E. x |^| F  =  { x } ) )
10 n0 3464 . . . . 5  |-  ( |^| F  =/=  (/)  <->  E. x  x  e. 
|^| F )
11 en1 6928 . . . . 5  |-  ( |^| F  ~~  1o  <->  E. x |^| F  =  { x } )
129, 10, 113imtr4g 261 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  ( |^| F  =/=  (/)  ->  |^| F  ~~  1o ) )
1312imp 418 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  |^| F  =/=  (/) )  ->  |^| F  ~~  1o )
14 endom 6888 . . 3  |-  ( |^| F  ~~  1o  ->  |^| F  ~<_  1o )
1513, 14syl 15 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  |^| F  =/=  (/) )  ->  |^| F  ~<_  1o )
16 1on 6486 . . 3  |-  1o  e.  On
17 0domg 6988 . . 3  |-  ( 1o  e.  On  ->  (/)  ~<_  1o )
1816, 17mp1i 11 . 2  |-  ( F  e.  ( UFil `  X
)  ->  (/)  ~<_  1o )
191, 15, 18pm2.61ne 2521 1  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  ~<_  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   {csn 3640   |^|cint 3862   class class class wbr 4023   Oncon0 4392   ` cfv 5255   1oc1o 6472    ~~ cen 6860    ~<_ cdom 6861   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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-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-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-we 4354  df-ord 4395  df-on 4396  df-suc 4398  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1o 6479  df-en 6864  df-dom 6865  df-fbas 17520  df-fg 17521  df-fil 17541  df-ufil 17596
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