MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ufildom1 Structured version   Unicode version

Theorem ufildom1 19615
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 4393 . 2  |-  ( |^| F  =  (/)  ->  ( |^| F  ~<_  1o  <->  (/)  ~<_  1o ) )
2 uffixsn 19614 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  { x }  e.  F )
3 intss1 4241 . . . . . . . . 9  |-  ( { x }  e.  F  ->  |^| F  C_  { x } )
42, 3syl 16 . . . . . . . 8  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  |^| F  C_  { x } )
5 simpr 461 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  x  e.  |^| F )
65snssd 4116 . . . . . . . 8  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  { x }  C_  |^| F )
74, 6eqssd 3471 . . . . . . 7  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  |^| F )  ->  |^| F  =  { x } )
87ex 434 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  |^| F  ->  |^| F  =  { x } ) )
98eximdv 1677 . . . . 5  |-  ( F  e.  ( UFil `  X
)  ->  ( E. x  x  e.  |^| F  ->  E. x |^| F  =  { x } ) )
10 n0 3744 . . . . 5  |-  ( |^| F  =/=  (/)  <->  E. x  x  e. 
|^| F )
11 en1 7476 . . . . 5  |-  ( |^| F  ~~  1o  <->  E. x |^| F  =  { x } )
129, 10, 113imtr4g 270 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  ( |^| F  =/=  (/)  ->  |^| F  ~~  1o ) )
1312imp 429 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  |^| F  =/=  (/) )  ->  |^| F  ~~  1o )
14 endom 7436 . . 3  |-  ( |^| F  ~~  1o  ->  |^| F  ~<_  1o )
1513, 14syl 16 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  |^| F  =/=  (/) )  ->  |^| F  ~<_  1o )
16 1on 7027 . . 3  |-  1o  e.  On
17 0domg 7538 . . 3  |-  ( 1o  e.  On  ->  (/)  ~<_  1o )
1816, 17mp1i 12 . 2  |-  ( F  e.  ( UFil `  X
)  ->  (/)  ~<_  1o )
191, 15, 18pm2.61ne 2763 1  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  ~<_  1o )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2644    C_ wss 3426   (/)c0 3735   {csn 3975   |^|cint 4226   class class class wbr 4390   Oncon0 4817   ` cfv 5516   1oc1o 7013    ~~ cen 7407    ~<_ cdom 7408   UFilcufil 19588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1o 7020  df-en 7411  df-dom 7412  df-fbas 17923  df-fg 17924  df-fil 19535  df-ufil 19590
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator