Theorem ufilmax 21711

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 1063	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → 𝐹 ⊆ 𝐺)
2		filelss 21656	. . . . . 6 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐺) → 𝑥 ⊆ 𝑋)
3	2	ex 450	. . . . 5 ⊢ (𝐺 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐺 → 𝑥 ⊆ 𝑋))
4	3	3ad2ant2 1083	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ∈ 𝐺 → 𝑥 ⊆ 𝑋))
5		ufilb 21710	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
6	5	3ad2antl1 1223	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
7		simpl3 1066	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ⊆ 𝐺)
8	7	sseld 3602	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → (𝑋 ∖ 𝑥) ∈ 𝐺))
9		filfbas 21652	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (Fil‘𝑋) → 𝐺 ∈ (fBas‘𝑋))
10		fbncp 21643	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ 𝐺) → ¬ (𝑋 ∖ 𝑥) ∈ 𝐺)
11	10	ex 450	. . . . . . . . . . . . 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 129	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐺 → ¬ 𝑥 ∈ 𝐺))
14	13	3ad2ant2 1083	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → ((𝑋 ∖ 𝑥) ∈ 𝐺 → ¬ 𝑥 ∈ 𝐺))
15	14	adantr 481	. . . . . . . . 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 230	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ 𝐺))
18	17	con4d 114	. . . . . 6 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐺 → 𝑥 ∈ 𝐹))
19	18	ex 450	. . . . 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 3609	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → 𝐺 ⊆ 𝐹)
23	1, 22	eqssd 3620	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → 𝐹 = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571   ⊆ wss 3574  ‘cfv 5888  fBascfbas 19734  Filcfil 21649  UFilcufil 21703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-fbas 19743  df-fil 21650  df-ufil 21705

This theorem is referenced by:  isufil2  21712  ufileu  21723  uffixfr  21727  fmufil  21763  uffclsflim  21835