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Theorem ufli 17609
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 17608 . . 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 3199 . . . 4  |-  ( g  =  F  ->  (
g  C_  f  <->  F  C_  f
) )
43rexbidv 2564 . . 3  |-  ( g  =  F  ->  ( E. f  e.  ( UFil `  X ) g 
C_  f  <->  E. f  e.  ( UFil `  X
) F  C_  f
) )
54rspccva 2883 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   ` cfv 5255   Filcfil 17540   UFilcufil 17594  UFLcufl 17595
This theorem is referenced by:  ssufl  17613  ufldom  17657  ufilcmp  17727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ufl 17597
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