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Theorem ufilmax 22203
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 1131 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐹𝐺)
2 filelss 22148 . . . . . 6 ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥𝐺) → 𝑥𝑋)
32ex 413 . . . . 5 (𝐺 ∈ (Fil‘𝑋) → (𝑥𝐺𝑥𝑋))
433ad2ant2 1127 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥𝐺𝑥𝑋))
5 ufilb 22202 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
653ad2antl1 1178 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
7 simpl3 1186 . . . . . . . . . 10 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → 𝐹𝐺)
87sseld 3894 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹 → (𝑋𝑥) ∈ 𝐺))
9 filfbas 22144 . . . . . . . . . . . . 13 (𝐺 ∈ (Fil‘𝑋) → 𝐺 ∈ (fBas‘𝑋))
10 fbncp 22135 . . . . . . . . . . . . . 14 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺) → ¬ (𝑋𝑥) ∈ 𝐺)
1110ex 413 . . . . . . . . . . . . 13 (𝐺 ∈ (fBas‘𝑋) → (𝑥𝐺 → ¬ (𝑋𝑥) ∈ 𝐺))
129, 11syl 17 . . . . . . . . . . . 12 (𝐺 ∈ (Fil‘𝑋) → (𝑥𝐺 → ¬ (𝑋𝑥) ∈ 𝐺))
1312con2d 136 . . . . . . . . . . 11 (𝐺 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ 𝐺 → ¬ 𝑥𝐺))
14133ad2ant2 1127 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → ((𝑋𝑥) ∈ 𝐺 → ¬ 𝑥𝐺))
1514adantr 481 . . . . . . . . 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 241 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 → ¬ 𝑥𝐺))
1817con4d 115 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥𝑋) → (𝑥𝐺𝑥𝐹))
1918ex 413 . . . . 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 3901 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐺𝐹)
231, 22eqssd 3912 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1525  wcel 2083  cdif 3862  wss 3865  cfv 6232  fBascfbas 20219  Filcfil 22141  UFilcufil 22195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fv 6240  df-fbas 20228  df-fil 22142  df-ufil 22197
This theorem is referenced by:  isufil2  22204  ufileu  22215  uffixfr  22219  fmufil  22255  uffclsflim  22327
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