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Theorem ufildom1 17950
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 4207 . 2  |-  ( |^| F  =  (/)  ->  ( |^| F  ~<_  1o  <->  (/)  ~<_  1o ) )
2 uffixsn 17949 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  { x }  e.  F )
3 intss1 4057 . . . . . . . . 9  |-  ( { x }  e.  F  ->  |^| F  C_  { x } )
42, 3syl 16 . . . . . . . 8  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  |^| F  C_  { x } )
5 simpr 448 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  x  e.  |^| F )
65snssd 3935 . . . . . . . 8  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  { x }  C_  |^| F )
74, 6eqssd 3357 . . . . . . 7  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  |^| F  =  { x } )
87ex 424 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  |^| F  ->  |^| F  =  { x } ) )
98eximdv 1632 . . . . 5  |-  ( F  e.  ( UFil `  X
)  ->  ( E. x  x  e.  |^| F  ->  E. x |^| F  =  { x } ) )
10 n0 3629 . . . . 5  |-  ( |^| F  =/=  (/)  <->  E. x  x  e. 
|^| F )
11 en1 7166 . . . . 5  |-  ( |^| F  ~~  1o  <->  E. x |^| F  =  { x } )
129, 10, 113imtr4g 262 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  ( |^| F  =/=  (/)  ->  |^| F  ~~  1o ) )
1312imp 419 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  |^| F  =/=  (/) )  ->  |^| F  ~~  1o )
14 endom 7126 . . 3  |-  ( |^| F  ~~  1o  ->  |^| F  ~<_  1o )
1513, 14syl 16 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  |^| F  =/=  (/) )  ->  |^| F  ~<_  1o )
16 1on 6723 . . 3  |-  1o  e.  On
17 0domg 7226 . . 3  |-  ( 1o  e.  On  ->  (/)  ~<_  1o )
1816, 17mp1i 12 . 2  |-  ( F  e.  ( UFil `  X
)  ->  (/)  ~<_  1o )
191, 15, 18pm2.61ne 2673 1  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  ~<_  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   {csn 3806   |^|cint 4042   class class class wbr 4204   Oncon0 4573   ` cfv 5446   1oc1o 6709    ~~ cen 7098    ~<_ cdom 7099   UFilcufil 17923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1o 6716  df-en 7102  df-dom 7103  df-fbas 16691  df-fg 16692  df-fil 17870  df-ufil 17925
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