Theorem zfbas 22028

Description: The set of upper sets of integers is a filter base on ℤ, which corresponds to convergence of sequences on ℤ. (Contributed by Mario Carneiro, 13-Oct-2015.)

Assertion

Ref	Expression
zfbas	⊢ ran ℤ≥ ∈ (fBas‘ℤ)

Proof of Theorem zfbas

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzf 11933	. . 3 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
2		frn 6262	. . 3 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ran ℤ≥ ⊆ 𝒫 ℤ)
3	1, 2	ax-mp 5	. 2 ⊢ ran ℤ≥ ⊆ 𝒫 ℤ
4		ffn 6256	. . . . . 6 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
5	1, 4	ax-mp 5	. . . . 5 ⊢ ℤ≥ Fn ℤ
6		1z 11697	. . . . 5 ⊢ 1 ∈ ℤ
7		fnfvelrn 6582	. . . . 5 ⊢ ((ℤ≥ Fn ℤ ∧ 1 ∈ ℤ) → (ℤ≥‘1) ∈ ran ℤ≥)
8	5, 6, 7	mp2an 684	. . . 4 ⊢ (ℤ≥‘1) ∈ ran ℤ≥
9	8	ne0ii 4124	. . 3 ⊢ ran ℤ≥ ≠ ∅
10		uzid 11945	. . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ (ℤ≥‘𝑥))
11		n0i 4120	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘𝑥) → ¬ (ℤ≥‘𝑥) = ∅)
12	10, 11	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℤ → ¬ (ℤ≥‘𝑥) = ∅)
13	12	nrex 3180	. . . . 5 ⊢ ¬ ∃𝑥 ∈ ℤ (ℤ≥‘𝑥) = ∅
14		fvelrnb 6468	. . . . . 6 ⊢ (ℤ≥ Fn ℤ → (∅ ∈ ran ℤ≥ ↔ ∃𝑥 ∈ ℤ (ℤ≥‘𝑥) = ∅))
15	5, 14	ax-mp 5	. . . . 5 ⊢ (∅ ∈ ran ℤ≥ ↔ ∃𝑥 ∈ ℤ (ℤ≥‘𝑥) = ∅)
16	13, 15	mtbir 315	. . . 4 ⊢ ¬ ∅ ∈ ran ℤ≥
17	16	nelir 3077	. . 3 ⊢ ∅ ∉ ran ℤ≥
18		uzin2 14425	. . . . 5 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) → (𝑥 ∩ 𝑦) ∈ ran ℤ≥)
19		vex 3388	. . . . . . 7 ⊢ 𝑥 ∈ V
20	19	inex1 4994	. . . . . 6 ⊢ (𝑥 ∩ 𝑦) ∈ V
21	20	pwid 4365	. . . . 5 ⊢ (𝑥 ∩ 𝑦) ∈ 𝒫 (𝑥 ∩ 𝑦)
22		inelcm 4227	. . . . 5 ⊢ (((𝑥 ∩ 𝑦) ∈ ran ℤ≥ ∧ (𝑥 ∩ 𝑦) ∈ 𝒫 (𝑥 ∩ 𝑦)) → (ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)
23	18, 21, 22	sylancl 581	. . . 4 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) → (ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)
24	23	rgen2a 3158	. . 3 ⊢ ∀𝑥 ∈ ran ℤ≥∀𝑦 ∈ ran ℤ≥(ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅
25	9, 17, 24	3pm3.2i 1439	. 2 ⊢ (ran ℤ≥ ≠ ∅ ∧ ∅ ∉ ran ℤ≥ ∧ ∀𝑥 ∈ ran ℤ≥∀𝑦 ∈ ran ℤ≥(ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)
26		zex 11675	. . 3 ⊢ ℤ ∈ V
27		isfbas 21961	. . 3 ⊢ (ℤ ∈ V → (ran ℤ≥ ∈ (fBas‘ℤ) ↔ (ran ℤ≥ ⊆ 𝒫 ℤ ∧ (ran ℤ≥ ≠ ∅ ∧ ∅ ∉ ran ℤ≥ ∧ ∀𝑥 ∈ ran ℤ≥∀𝑦 ∈ ran ℤ≥(ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))))
28	26, 27	ax-mp 5	. 2 ⊢ (ran ℤ≥ ∈ (fBas‘ℤ) ↔ (ran ℤ≥ ⊆ 𝒫 ℤ ∧ (ran ℤ≥ ≠ ∅ ∧ ∅ ∉ ran ℤ≥ ∧ ∀𝑥 ∈ ran ℤ≥∀𝑦 ∈ ran ℤ≥(ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))
29	3, 25, 28	mpbir2an 703	1 ⊢ ran ℤ≥ ∈ (fBas‘ℤ)

Syntax hints:  ¬ wn 3   ↔ wb 198   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157   ≠ wne 2971   ∉ wnel 3074  ∀wral 3089  ∃wrex 3090  Vcvv 3385   ∩ cin 3768   ⊆ wss 3769  ∅c0 4115  𝒫 cpw 4349  ran crn 5313   Fn wfn 6096  ⟶wf 6097  ‘cfv 6101  1c1 10225  ℤcz 11666  ℤ_≥cuz 11930  fBascfbas 20056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-i2m1 10292  ax-1ne0 10293  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-neg 10559  df-nn 11313  df-z 11667  df-uz 11931  df-fbas 20065

This theorem is referenced by:  uzfbas  22030