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Theorem ufilmax 23831
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 1135 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐹𝐺)
2 filelss 23776 . . . . . 6 ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥𝐺) → 𝑥𝑋)
32ex 411 . . . . 5 (𝐺 ∈ (Fil‘𝑋) → (𝑥𝐺𝑥𝑋))
433ad2ant2 1131 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥𝐺𝑥𝑋))
5 ufilb 23830 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
653ad2antl1 1182 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
7 simpl3 1190 . . . . . . . . . 10 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → 𝐹𝐺)
87sseld 3981 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹 → (𝑋𝑥) ∈ 𝐺))
9 filfbas 23772 . . . . . . . . . . . . 13 (𝐺 ∈ (Fil‘𝑋) → 𝐺 ∈ (fBas‘𝑋))
10 fbncp 23763 . . . . . . . . . . . . . 14 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺) → ¬ (𝑋𝑥) ∈ 𝐺)
1110ex 411 . . . . . . . . . . . . 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 1131 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → ((𝑋𝑥) ∈ 𝐺 → ¬ 𝑥𝐺))
1514adantr 479 . . . . . . . . 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 239 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 → ¬ 𝑥𝐺))
1817con4d 115 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (𝑥𝐺𝑥𝐹))
1918ex 411 . . . . 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 3988 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐺𝐹)
231, 22eqssd 3999 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  cdif 3946  wss 3949  cfv 6553  fBascfbas 21274  Filcfil 23769  UFilcufil 23823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-fbas 21283  df-fil 23770  df-ufil 23825
This theorem is referenced by:  isufil2  23832  ufileu  23843  uffixfr  23847  fmufil  23883  uffclsflim  23955
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