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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 12234 | . . 3 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
2 | frn 6493 | . . 3 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ran ℤ≥ ⊆ 𝒫 ℤ) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ran ℤ≥ ⊆ 𝒫 ℤ |
4 | ffn 6487 | . . . . . 6 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ ℤ≥ Fn ℤ |
6 | 1z 12000 | . . . . 5 ⊢ 1 ∈ ℤ | |
7 | fnfvelrn 6825 | . . . . 5 ⊢ ((ℤ≥ Fn ℤ ∧ 1 ∈ ℤ) → (ℤ≥‘1) ∈ ran ℤ≥) | |
8 | 5, 6, 7 | mp2an 691 | . . . 4 ⊢ (ℤ≥‘1) ∈ ran ℤ≥ |
9 | 8 | ne0ii 4253 | . . 3 ⊢ ran ℤ≥ ≠ ∅ |
10 | uzid 12246 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ (ℤ≥‘𝑥)) | |
11 | n0i 4249 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘𝑥) → ¬ (ℤ≥‘𝑥) = ∅) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → ¬ (ℤ≥‘𝑥) = ∅) |
13 | 12 | nrex 3228 | . . . . 5 ⊢ ¬ ∃𝑥 ∈ ℤ (ℤ≥‘𝑥) = ∅ |
14 | fvelrnb 6701 | . . . . . 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 326 | . . . 4 ⊢ ¬ ∅ ∈ ran ℤ≥ |
17 | 16 | nelir 3094 | . . 3 ⊢ ∅ ∉ ran ℤ≥ |
18 | uzin2 14696 | . . . . 5 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) → (𝑥 ∩ 𝑦) ∈ ran ℤ≥) | |
19 | vex 3444 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
20 | 19 | inex1 5185 | . . . . . 6 ⊢ (𝑥 ∩ 𝑦) ∈ V |
21 | 20 | pwid 4521 | . . . . 5 ⊢ (𝑥 ∩ 𝑦) ∈ 𝒫 (𝑥 ∩ 𝑦) |
22 | inelcm 4372 | . . . . 5 ⊢ (((𝑥 ∩ 𝑦) ∈ ran ℤ≥ ∧ (𝑥 ∩ 𝑦) ∈ 𝒫 (𝑥 ∩ 𝑦)) → (ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅) | |
23 | 18, 21, 22 | sylancl 589 | . . . 4 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) → (ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅) |
24 | 23 | rgen2 3168 | . . 3 ⊢ ∀𝑥 ∈ ran ℤ≥∀𝑦 ∈ ran ℤ≥(ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ |
25 | 9, 17, 24 | 3pm3.2i 1336 | . 2 ⊢ (ran ℤ≥ ≠ ∅ ∧ ∅ ∉ ran ℤ≥ ∧ ∀𝑥 ∈ ran ℤ≥∀𝑦 ∈ ran ℤ≥(ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅) |
26 | zex 11978 | . . 3 ⊢ ℤ ∈ V | |
27 | isfbas 22434 | . . 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 710 | 1 ⊢ ran ℤ≥ ∈ (fBas‘ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∉ wnel 3091 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 ran crn 5520 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 1c1 10527 ℤcz 11969 ℤ≥cuz 12231 fBascfbas 20079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-i2m1 10594 ax-1ne0 10595 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-nn 11626 df-z 11970 df-uz 12232 df-fbas 20088 |
This theorem is referenced by: uzfbas 22503 |
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