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Theorem ufli 17948
Description: Property of a set that satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
ufli  |-  ( ( X  e. UFL  /\  F  e.  ( Fil `  X
) )  ->  E. f  e.  ( UFil `  X
) F  C_  f
)
Distinct variable groups:    f, F    f, X

Proof of Theorem ufli
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 isufl 17947 . . 3  |-  ( X  e. UFL  ->  ( X  e. UFL  <->  A. g  e.  ( Fil `  X ) E. f  e.  ( UFil `  X
) g  C_  f
) )
21ibi 234 . 2  |-  ( X  e. UFL  ->  A. g  e.  ( Fil `  X ) E. f  e.  (
UFil `  X )
g  C_  f )
3 sseq1 3371 . . . 4  |-  ( g  =  F  ->  (
g  C_  f  <->  F  C_  f
) )
43rexbidv 2728 . . 3  |-  ( g  =  F  ->  ( E. f  e.  ( UFil `  X ) g 
C_  f  <->  E. f  e.  ( UFil `  X
) F  C_  f
) )
54rspccva 3053 . 2  |-  ( ( A. g  e.  ( Fil `  X ) E. f  e.  (
UFil `  X )
g  C_  f  /\  F  e.  ( Fil `  X ) )  ->  E. f  e.  ( UFil `  X ) F 
C_  f )
62, 5sylan 459 1  |-  ( ( X  e. UFL  /\  F  e.  ( Fil `  X
) )  ->  E. f  e.  ( UFil `  X
) F  C_  f
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    C_ wss 3322   ` cfv 5456   Filcfil 17879   UFilcufil 17933  UFLcufl 17934
This theorem is referenced by:  ssufl  17952  ufldom  17996  ufilcmp  18066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ufl 17936
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