Theorem ufilfil 24224

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 24223	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))
2	1	simplbi 502	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   ∈ wcel 2144  ∀wral 3076   ∖ cdif 4045  𝒫 cpw 4556  ‘cfv 6538  Filcfil 24165  UFilcufil 24219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fv 6546  df-ufil 24221

This theorem is used by:  ufilb  24226  isufil2  24228  ufprim  24229  trufil  24230  ufileu  24239  filufint  24240  uffixfr  24243  uffix2  24244  uffixsn  24245  uffinfix  24247  cfinufil  24248  ufilen  24250  ufildr  24251  fmufil  24279  ufldom  24282  uffclsflim  24351  ufilcmp  24352  uffcfflf  24359  alexsublem  24364