MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ufilmax Structured version   Visualization version   GIF version

Theorem ufilmax 21924
Description: Any filter finer than an ultrafilter is actually equal to it. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
Assertion
Ref Expression
ufilmax ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐹 = 𝐺)

Proof of Theorem ufilmax
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp3 1161 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐹𝐺)
2 filelss 21869 . . . . . 6 ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥𝐺) → 𝑥𝑋)
32ex 399 . . . . 5 (𝐺 ∈ (Fil‘𝑋) → (𝑥𝐺𝑥𝑋))
433ad2ant2 1157 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥𝐺𝑥𝑋))
5 ufilb 21923 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
653ad2antl1 1229 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
7 simpl3 1239 . . . . . . . . . 10 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → 𝐹𝐺)
87sseld 3797 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹 → (𝑋𝑥) ∈ 𝐺))
9 filfbas 21865 . . . . . . . . . . . . 13 (𝐺 ∈ (Fil‘𝑋) → 𝐺 ∈ (fBas‘𝑋))
10 fbncp 21856 . . . . . . . . . . . . . 14 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺) → ¬ (𝑋𝑥) ∈ 𝐺)
1110ex 399 . . . . . . . . . . . . 13 (𝐺 ∈ (fBas‘𝑋) → (𝑥𝐺 → ¬ (𝑋𝑥) ∈ 𝐺))
129, 11syl 17 . . . . . . . . . . . 12 (𝐺 ∈ (Fil‘𝑋) → (𝑥𝐺 → ¬ (𝑋𝑥) ∈ 𝐺))
1312con2d 131 . . . . . . . . . . 11 (𝐺 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ 𝐺 → ¬ 𝑥𝐺))
14133ad2ant2 1157 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → ((𝑋𝑥) ∈ 𝐺 → ¬ 𝑥𝐺))
1514adantr 468 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐺 → ¬ 𝑥𝐺))
168, 15syld 47 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹 → ¬ 𝑥𝐺))
176, 16sylbid 231 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 → ¬ 𝑥𝐺))
1817con4d 115 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (𝑥𝐺𝑥𝐹))
1918ex 399 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥𝑋 → (𝑥𝐺𝑥𝐹)))
2019com23 86 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥𝐺 → (𝑥𝑋𝑥𝐹)))
214, 20mpdd 43 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥𝐺𝑥𝐹))
2221ssrdv 3804 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐺𝐹)
231, 22eqssd 3815 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  cdif 3766  wss 3769  cfv 6101  fBascfbas 19942  Filcfil 21862  UFilcufil 21916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fv 6109  df-fbas 19951  df-fil 21863  df-ufil 21918
This theorem is referenced by:  isufil2  21925  ufileu  21936  uffixfr  21940  fmufil  21976  uffclsflim  22048
  Copyright terms: Public domain W3C validator