Theorem ufilmax 24133

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 1156	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → 𝐹 ⊆ 𝐺)
2		filelss 24078	. . . . . 6 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐺) → 𝑥 ⊆ 𝑋)
3	2	ex 418	. . . . 5 ⊢ (𝐺 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐺 → 𝑥 ⊆ 𝑋))
4	3	3ad2ant2 1152	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ∈ 𝐺 → 𝑥 ⊆ 𝑋))
5		ufilb 24132	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
6	5	3ad2antl1 1204	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
7		simpl3 1212	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ⊆ 𝐺)
8	7	sseld 3930	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → (𝑋 ∖ 𝑥) ∈ 𝐺))
9		filfbas 24074	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (Fil‘𝑋) → 𝐺 ∈ (fBas‘𝑋))
10		fbncp 24065	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ 𝐺) → ¬ (𝑋 ∖ 𝑥) ∈ 𝐺)
11	10	ex 418	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ 𝐺 → ¬ (𝑋 ∖ 𝑥) ∈ 𝐺))
12	9, 11	syl 18	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐺 → ¬ (𝑋 ∖ 𝑥) ∈ 𝐺))
13	12	con2d 135	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐺 → ¬ 𝑥 ∈ 𝐺))
14	13	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → ((𝑋 ∖ 𝑥) ∈ 𝐺 → ¬ 𝑥 ∈ 𝐺))
15	14	adantr 486	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐺 → ¬ 𝑥 ∈ 𝐺))
16	8, 15	syld 48	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ¬ 𝑥 ∈ 𝐺))
17	6, 16	sylbid 243	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ 𝐺))
18	17	con4d 116	. . . . . 6 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐺 → 𝑥 ∈ 𝐹))
19	18	ex 418	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ⊆ 𝑋 → (𝑥 ∈ 𝐺 → 𝑥 ∈ 𝐹)))
20	19	com23 87	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ∈ 𝐺 → (𝑥 ⊆ 𝑋 → 𝑥 ∈ 𝐹)))
21	4, 20	mpdd 44	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ∈ 𝐺 → 𝑥 ∈ 𝐹))
22	21	ssrdv 3937	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → 𝐺 ⊆ 𝐹)
23	1, 22	eqssd 3948	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → 𝐹 = 𝐺)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896   ⊆ wss 3899  ‘cfv 6533  fBascfbas 21573  Filcfil 24071  UFilcufil 24125

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fv 6541  df-fbas 21582  df-fil 24072  df-ufil 24127

This theorem is used by:  isufil2  24134  ufileu  24145  uffixfr  24149  fmufil  24185  uffclsflim  24257