Theorem ufilfil 23791

Description: An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

Assertion

Ref	Expression
ufilfil	⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))

Proof of Theorem ufilfil

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isufil 23790	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))
2	1	simplbi 497	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ∨ wo 847   ∈ wcel 2109  ∀wral 3044   ∖ cdif 3911  𝒫 cpw 4563  ‘cfv 6511  Filcfil 23732  UFilcufil 23786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ufil 23788

This theorem is referenced by:  ufilb  23793  isufil2  23795  ufprim  23796  trufil  23797  ufileu  23806  filufint  23807  uffixfr  23810  uffix2  23811  uffixsn  23812  uffinfix  23814  cfinufil  23815  ufilen  23817  ufildr  23818  fmufil  23846  ufldom  23849  uffclsflim  23918  ufilcmp  23919  uffcfflf  23926  alexsublem  23931