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Theorem isufil 19618
Description: The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
Assertion
Ref Expression
isufil  |-  ( F  e.  ( UFil `  X
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
Distinct variable groups:    x, F    x, X

Proof of Theorem isufil
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ufil 19616 . 2  |-  UFil  =  ( y  e.  _V  |->  { z  e.  ( Fil `  y )  |  A. x  e. 
~P  y ( x  e.  z  \/  (
y  \  x )  e.  z ) } )
2 pweq 3974 . . . 4  |-  ( y  =  X  ->  ~P y  =  ~P X
)
32adantr 465 . . 3  |-  ( ( y  =  X  /\  z  =  F )  ->  ~P y  =  ~P X )
4 eleq2 2527 . . . . 5  |-  ( z  =  F  ->  (
x  e.  z  <->  x  e.  F ) )
54adantl 466 . . . 4  |-  ( ( y  =  X  /\  z  =  F )  ->  ( x  e.  z  <-> 
x  e.  F ) )
6 difeq1 3578 . . . . 5  |-  ( y  =  X  ->  (
y  \  x )  =  ( X  \  x ) )
7 eleq12 2530 . . . . 5  |-  ( ( ( y  \  x
)  =  ( X 
\  x )  /\  z  =  F )  ->  ( ( y  \  x )  e.  z  <-> 
( X  \  x
)  e.  F ) )
86, 7sylan 471 . . . 4  |-  ( ( y  =  X  /\  z  =  F )  ->  ( ( y  \  x )  e.  z  <-> 
( X  \  x
)  e.  F ) )
95, 8orbi12d 709 . . 3  |-  ( ( y  =  X  /\  z  =  F )  ->  ( ( x  e.  z  \/  ( y 
\  x )  e.  z )  <->  ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
103, 9raleqbidv 3037 . 2  |-  ( ( y  =  X  /\  z  =  F )  ->  ( A. x  e. 
~P  y ( x  e.  z  \/  (
y  \  x )  e.  z )  <->  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
11 fveq2 5802 . 2  |-  ( y  =  X  ->  ( Fil `  y )  =  ( Fil `  X
) )
12 fvex 5812 . 2  |-  ( Fil `  y )  e.  _V
13 elfvdm 5828 . 2  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  dom  Fil )
141, 10, 11, 12, 13elmptrab2 19543 1  |-  ( F  e.  ( UFil `  X
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078    \ cdif 3436   ~Pcpw 3971   dom cdm 4951   ` cfv 5529   Filcfil 19560   UFilcufil 19614
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fv 5537  df-ufil 19616
This theorem is referenced by:  ufilfil  19619  ufilss  19620  isufil2  19623  trufil  19625  fixufil  19637  fin1aufil  19647
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