Theorem ufilfil 24072

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 24071	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))
2	1	simplbi 501	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∨ wo 860   ∈ wcel 2142  ∀wral 3078   ∖ cdif 3901  𝒫 cpw 4561  ‘cfv 6536  Filcfil 24013  UFilcufil 24067

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-ufil 24069

This theorem is used by:  ufilb  24074  isufil2  24076  ufprim  24077  trufil  24078  ufileu  24087  filufint  24088  uffixfr  24091  uffix2  24092  uffixsn  24093  uffinfix  24095  cfinufil  24096  ufilen  24098  ufildr  24099  fmufil  24127  ufldom  24130  uffclsflim  24199  ufilcmp  24200  uffcfflf  24207  alexsublem  24212