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

Theorem ufli 20541
Description: Property of a set that satisfies 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 20540 . . 3  |-  ( X  e. UFL  ->  ( X  e. UFL  <->  A. g  e.  ( Fil `  X ) E. f  e.  ( UFil `  X
) g  C_  f
) )
21ibi 241 . 2  |-  ( X  e. UFL  ->  A. g  e.  ( Fil `  X ) E. f  e.  (
UFil `  X )
g  C_  f )
3 sseq1 3520 . . . 4  |-  ( g  =  F  ->  (
g  C_  f  <->  F  C_  f
) )
43rexbidv 2968 . . 3  |-  ( g  =  F  ->  ( E. f  e.  ( UFil `  X ) g 
C_  f  <->  E. f  e.  ( UFil `  X
) F  C_  f
) )
54rspccva 3209 . 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 471 1  |-  ( ( X  e. UFL  /\  F  e.  ( Fil `  X
) )  ->  E. f  e.  ( UFil `  X
) F  C_  f
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    C_ wss 3471   ` cfv 5594   Filcfil 20472   UFilcufil 20526  UFLcufl 20527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ufl 20529
This theorem is referenced by:  ssufl  20545  ufldom  20589  ufilcmp  20659
  Copyright terms: Public domain W3C validator