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

Theorem ufilmax 23915
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 1139 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐹𝐺)
2 filelss 23860 . . . . . 6 ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥𝐺) → 𝑥𝑋)
32ex 412 . . . . 5 (𝐺 ∈ (Fil‘𝑋) → (𝑥𝐺𝑥𝑋))
433ad2ant2 1135 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥𝐺𝑥𝑋))
5 ufilb 23914 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
653ad2antl1 1186 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
7 simpl3 1194 . . . . . . . . . 10 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → 𝐹𝐺)
87sseld 3982 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹 → (𝑋𝑥) ∈ 𝐺))
9 filfbas 23856 . . . . . . . . . . . . 13 (𝐺 ∈ (Fil‘𝑋) → 𝐺 ∈ (fBas‘𝑋))
10 fbncp 23847 . . . . . . . . . . . . . 14 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺) → ¬ (𝑋𝑥) ∈ 𝐺)
1110ex 412 . . . . . . . . . . . . 13 (𝐺 ∈ (fBas‘𝑋) → (𝑥𝐺 → ¬ (𝑋𝑥) ∈ 𝐺))
129, 11syl 17 . . . . . . . . . . . 12 (𝐺 ∈ (Fil‘𝑋) → (𝑥𝐺 → ¬ (𝑋𝑥) ∈ 𝐺))
1312con2d 134 . . . . . . . . . . 11 (𝐺 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ 𝐺 → ¬ 𝑥𝐺))
14133ad2ant2 1135 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → ((𝑋𝑥) ∈ 𝐺 → ¬ 𝑥𝐺))
1514adantr 480 . . . . . . . . 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 240 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 → ¬ 𝑥𝐺))
1817con4d 115 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (𝑥𝐺𝑥𝐹))
1918ex 412 . . . . 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 3989 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐺𝐹)
231, 22eqssd 4001 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cdif 3948  wss 3951  cfv 6561  fBascfbas 21352  Filcfil 23853  UFilcufil 23907
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-fbas 21361  df-fil 23854  df-ufil 23909
This theorem is referenced by:  isufil2  23916  ufileu  23927  uffixfr  23931  fmufil  23967  uffclsflim  24039
  Copyright terms: Public domain W3C validator