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Mirrors > Home > MPE Home > Th. List > zfbas | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
zfbas | ⊢ ran ℤ≥ ∈ (fBas‘ℤ) |
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) |
Colors of variables: wff setvar class |
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 |
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