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) ∧ 𝐹 ⊆ 𝐺) → 𝐹 = 𝐺)

Proof of Theorem ufilmax

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1139	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → 𝐹 ⊆ 𝐺)
2		filelss 23860	. . . . . 6 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐺) → 𝑥 ⊆ 𝑋)
3	2	ex 412	. . . . 5 ⊢ (𝐺 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐺 → 𝑥 ⊆ 𝑋))
4	3	3ad2ant2 1135	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ∈ 𝐺 → 𝑥 ⊆ 𝑋))
5		ufilb 23914	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
6	5	3ad2antl1 1186	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
7		simpl3 1194	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ⊆ 𝐺)
8	7	sseld 3982	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → (𝑋 ∖ 𝑥) ∈ 𝐺))
9		filfbas 23856	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (Fil‘𝑋) → 𝐺 ∈ (fBas‘𝑋))
10		fbncp 23847	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ 𝐺) → ¬ (𝑋 ∖ 𝑥) ∈ 𝐺)
11	10	ex 412	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ 𝐺 → ¬ (𝑋 ∖ 𝑥) ∈ 𝐺))
12	9, 11	syl 17	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐺 → ¬ (𝑋 ∖ 𝑥) ∈ 𝐺))
13	12	con2d 134	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐺 → ¬ 𝑥 ∈ 𝐺))
14	13	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → ((𝑋 ∖ 𝑥) ∈ 𝐺 → ¬ 𝑥 ∈ 𝐺))
15	14	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐺 → ¬ 𝑥 ∈ 𝐺))
16	8, 15	syld 47	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ¬ 𝑥 ∈ 𝐺))
17	6, 16	sylbid 240	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ 𝐺))
18	17	con4d 115	. . . . . 6 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐺 → 𝑥 ∈ 𝐹))
19	18	ex 412	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ⊆ 𝑋 → (𝑥 ∈ 𝐺 → 𝑥 ∈ 𝐹)))
20	19	com23 86	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ∈ 𝐺 → (𝑥 ⊆ 𝑋 → 𝑥 ∈ 𝐹)))
21	4, 20	mpdd 43	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ∈ 𝐺 → 𝑥 ∈ 𝐹))
22	21	ssrdv 3989	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → 𝐺 ⊆ 𝐹)
23	1, 22	eqssd 4001	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → 𝐹 = 𝐺)

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