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Theorem ufilmax 21463
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 1055 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐹𝐺)
2 filelss 21408 . . . . . 6 ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥𝐺) → 𝑥𝑋)
32ex 448 . . . . 5 (𝐺 ∈ (Fil‘𝑋) → (𝑥𝐺𝑥𝑋))
433ad2ant2 1075 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥𝐺𝑥𝑋))
5 ufilb 21462 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
653ad2antl1 1215 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
7 simpl3 1058 . . . . . . . . . 10 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → 𝐹𝐺)
87sseld 3566 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹 → (𝑋𝑥) ∈ 𝐺))
9 filfbas 21404 . . . . . . . . . . . . 13 (𝐺 ∈ (Fil‘𝑋) → 𝐺 ∈ (fBas‘𝑋))
10 fbncp 21395 . . . . . . . . . . . . . 14 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺) → ¬ (𝑋𝑥) ∈ 𝐺)
1110ex 448 . . . . . . . . . . . . 13 (𝐺 ∈ (fBas‘𝑋) → (𝑥𝐺 → ¬ (𝑋𝑥) ∈ 𝐺))
129, 11syl 17 . . . . . . . . . . . 12 (𝐺 ∈ (Fil‘𝑋) → (𝑥𝐺 → ¬ (𝑋𝑥) ∈ 𝐺))
1312con2d 127 . . . . . . . . . . 11 (𝐺 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ 𝐺 → ¬ 𝑥𝐺))
14133ad2ant2 1075 . . . . . . . . . 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 45 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹 → ¬ 𝑥𝐺))
176, 16sylbid 228 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 → ¬ 𝑥𝐺))
1817con4d 112 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (𝑥𝐺𝑥𝐹))
1918ex 448 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥𝑋 → (𝑥𝐺𝑥𝐹)))
2019com23 83 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥𝐺 → (𝑥𝑋𝑥𝐹)))
214, 20mpdd 41 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥𝐺𝑥𝐹))
2221ssrdv 3573 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐺𝐹)
231, 22eqssd 3584 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  cdif 3536  wss 3539  cfv 5790  fBascfbas 19501  Filcfil 21401  UFilcufil 21455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fv 5798  df-fbas 19510  df-fil 21402  df-ufil 21457
This theorem is referenced by:  isufil2  21464  ufileu  21475  uffixfr  21479  fmufil  21515  uffclsflim  21587
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