Theorem ufilmax 23794

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 1138	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → 𝐹 ⊆ 𝐺)
2		filelss 23739	. . . . . 6 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐺) → 𝑥 ⊆ 𝑋)
3	2	ex 412	. . . . 5 ⊢ (𝐺 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐺 → 𝑥 ⊆ 𝑋))
4	3	3ad2ant2 1134	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ∈ 𝐺 → 𝑥 ⊆ 𝑋))
5		ufilb 23793	. . . . . . . . 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 3945	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → (𝑋 ∖ 𝑥) ∈ 𝐺))
9		filfbas 23735	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (Fil‘𝑋) → 𝐺 ∈ (fBas‘𝑋))
10		fbncp 23726	. . . . . . . . . . . . . 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 1134	. . . . . . . . . 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 3952	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → 𝐺 ⊆ 𝐹)
23	1, 22	eqssd 3964	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → 𝐹 = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∖ cdif 3911   ⊆ wss 3914  ‘cfv 6511  fBascfbas 21252  Filcfil 23732  UFilcufil 23786

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-fbas 21261  df-fil 23733  df-ufil 23788

This theorem is referenced by:  isufil2  23795  ufileu  23806  uffixfr  23810  fmufil  23846  uffclsflim  23918