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Theorem ufli 17625
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 17624 . . 3  |-  ( X  e. UFL  ->  ( X  e. UFL  <->  A. g  e.  ( Fil `  X ) E. f  e.  ( UFil `  X
) g  C_  f
) )
21ibi 232 . 2  |-  ( X  e. UFL  ->  A. g  e.  ( Fil `  X ) E. f  e.  (
UFil `  X )
g  C_  f )
3 sseq1 3212 . . . 4  |-  ( g  =  F  ->  (
g  C_  f  <->  F  C_  f
) )
43rexbidv 2577 . . 3  |-  ( g  =  F  ->  ( E. f  e.  ( UFil `  X ) g 
C_  f  <->  E. f  e.  ( UFil `  X
) F  C_  f
) )
54rspccva 2896 . 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 457 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 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   ` cfv 5271   Filcfil 17556   UFilcufil 17610  UFLcufl 17611
This theorem is referenced by:  ssufl  17629  ufldom  17673  ufilcmp  17743
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ufl 17613
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