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Theorem ufildom1 20996
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 4421 . 2  |-  ( |^| F  =  (/)  ->  ( |^| F  ~<_  1o  <->  (/)  ~<_  1o ) )
2 uffixsn 20995 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  { x }  e.  F )
3 intss1 4263 . . . . . . . . 9  |-  ( { x }  e.  F  ->  |^| F  C_  { x } )
42, 3syl 17 . . . . . . . 8  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  |^| F  C_  { x } )
5 simpr 467 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  x  e.  |^| F )
65snssd 4130 . . . . . . . 8  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  { x }  C_  |^| F )
74, 6eqssd 3461 . . . . . . 7  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  |^| F  =  { x } )
87ex 440 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  |^| F  ->  |^| F  =  { x } ) )
98eximdv 1775 . . . . 5  |-  ( F  e.  ( UFil `  X
)  ->  ( E. x  x  e.  |^| F  ->  E. x |^| F  =  { x } ) )
10 n0 3753 . . . . 5  |-  ( |^| F  =/=  (/)  <->  E. x  x  e. 
|^| F )
11 en1 7667 . . . . 5  |-  ( |^| F  ~~  1o  <->  E. x |^| F  =  { x } )
129, 10, 113imtr4g 278 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  ( |^| F  =/=  (/)  ->  |^| F  ~~  1o ) )
1312imp 435 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  |^| F  =/=  (/) )  ->  |^| F  ~~  1o )
14 endom 7627 . . 3  |-  ( |^| F  ~~  1o  ->  |^| F  ~<_  1o )
1513, 14syl 17 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  |^| F  =/=  (/) )  ->  |^| F  ~<_  1o )
16 1on 7220 . . 3  |-  1o  e.  On
17 0domg 7730 . . 3  |-  ( 1o  e.  On  ->  (/)  ~<_  1o )
1816, 17mp1i 13 . 2  |-  ( F  e.  ( UFil `  X
)  ->  (/)  ~<_  1o )
191, 15, 18pm2.61ne 2721 1  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  ~<_  1o )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455   E.wex 1674    e. wcel 1898    =/= wne 2633    C_ wss 3416   (/)c0 3743   {csn 3980   |^|cint 4248   class class class wbr 4418   Oncon0 5446   ` cfv 5605   1oc1o 7206    ~~ cen 7597    ~<_ cdom 7598   UFilcufil 20969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-ord 5449  df-on 5450  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-1o 7213  df-en 7601  df-dom 7602  df-fbas 19022  df-fg 19023  df-fil 20916  df-ufil 20971
This theorem is referenced by: (None)
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