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Theorem filelss 22033
Description: An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
filelss ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)

Proof of Theorem filelss
StepHypRef Expression
1 filfbas 22029 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 fbelss 22014 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)
31, 2sylan 575 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2164  wss 3798  cfv 6127  fBascfbas 20101  Filcfil 22026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fv 6135  df-fbas 20110  df-fil 22027
This theorem is referenced by:  filin  22035  filtop  22036  filuni  22066  trfil2  22068  trfil3  22069  fgtr  22071  trfg  22072  ufilmax  22088  isufil2  22089  ufileu  22100  filufint  22101  cfinufil  22109  ufilen  22111  rnelfm  22134  fmfnfmlem4  22138  fmid  22141  flimclsi  22159  flimrest  22164  txflf  22187  fclsopn  22195  fclsrest  22205  flimfnfcls  22209  fclscmpi  22210  iscfil2  23441  cfil3i  23444  iscmet3lem2  23467  iscmet3  23468  cfilresi  23470  cfilres  23471  filnetlem3  32908
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