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Theorem ufildom1 17637
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 4042 . 2  |-  ( |^| F  =  (/)  ->  ( |^| F  ~<_  1o  <->  (/)  ~<_  1o ) )
2 uffixsn 17636 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  { x }  e.  F )
3 intss1 3893 . . . . . . . . 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 3776 . . . . . . . 8  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  { x }  C_  |^| F )
74, 6eqssd 3209 . . . . . . 7  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  |^| F  =  { x } )
87ex 423 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  |^| F  ->  |^| F  =  { x } ) )
98eximdv 1612 . . . . 5  |-  ( F  e.  ( UFil `  X
)  ->  ( E. x  x  e.  |^| F  ->  E. x |^| F  =  { x } ) )
10 n0 3477 . . . . 5  |-  ( |^| F  =/=  (/)  <->  E. x  x  e. 
|^| F )
11 en1 6944 . . . . 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 6904 . . 3  |-  ( |^| F  ~~  1o  ->  |^| F  ~<_  1o )
1513, 14syl 15 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  |^| F  =/=  (/) )  ->  |^| F  ~<_  1o )
16 1on 6502 . . 3  |-  1o  e.  On
17 0domg 7004 . . 3  |-  ( 1o  e.  On  ->  (/)  ~<_  1o )
1816, 17mp1i 11 . 2  |-  ( F  e.  ( UFil `  X
)  ->  (/)  ~<_  1o )
191, 15, 18pm2.61ne 2534 1  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  ~<_  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   {csn 3653   |^|cint 3878   class class class wbr 4039   Oncon0 4408   ` cfv 5271   1oc1o 6488    ~~ cen 6876    ~<_ cdom 6877   UFilcufil 17610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1o 6495  df-en 6880  df-dom 6881  df-fbas 17536  df-fg 17537  df-fil 17557  df-ufil 17612
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