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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 12858 | . . 3 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
2 | frn 6730 | . . 3 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ran ℤ≥ ⊆ 𝒫 ℤ) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ran ℤ≥ ⊆ 𝒫 ℤ |
4 | ffn 6723 | . . . . . 6 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ ℤ≥ Fn ℤ |
6 | 1z 12625 | . . . . 5 ⊢ 1 ∈ ℤ | |
7 | fnfvelrn 7089 | . . . . 5 ⊢ ((ℤ≥ Fn ℤ ∧ 1 ∈ ℤ) → (ℤ≥‘1) ∈ ran ℤ≥) | |
8 | 5, 6, 7 | mp2an 690 | . . . 4 ⊢ (ℤ≥‘1) ∈ ran ℤ≥ |
9 | 8 | ne0ii 4337 | . . 3 ⊢ ran ℤ≥ ≠ ∅ |
10 | uzid 12870 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ (ℤ≥‘𝑥)) | |
11 | n0i 4333 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘𝑥) → ¬ (ℤ≥‘𝑥) = ∅) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → ¬ (ℤ≥‘𝑥) = ∅) |
13 | 12 | nrex 3063 | . . . . 5 ⊢ ¬ ∃𝑥 ∈ ℤ (ℤ≥‘𝑥) = ∅ |
14 | fvelrnb 6958 | . . . . . 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 322 | . . . 4 ⊢ ¬ ∅ ∈ ran ℤ≥ |
17 | 16 | nelir 3038 | . . 3 ⊢ ∅ ∉ ran ℤ≥ |
18 | uzin2 15327 | . . . . 5 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) → (𝑥 ∩ 𝑦) ∈ ran ℤ≥) | |
19 | vex 3465 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
20 | 19 | inex1 5318 | . . . . . 6 ⊢ (𝑥 ∩ 𝑦) ∈ V |
21 | 20 | pwid 4626 | . . . . 5 ⊢ (𝑥 ∩ 𝑦) ∈ 𝒫 (𝑥 ∩ 𝑦) |
22 | inelcm 4466 | . . . . 5 ⊢ (((𝑥 ∩ 𝑦) ∈ ran ℤ≥ ∧ (𝑥 ∩ 𝑦) ∈ 𝒫 (𝑥 ∩ 𝑦)) → (ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅) | |
23 | 18, 21, 22 | sylancl 584 | . . . 4 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) → (ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅) |
24 | 23 | rgen2 3187 | . . 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 12600 | . . 3 ⊢ ℤ ∈ V | |
27 | isfbas 23777 | . . 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 709 | 1 ⊢ ran ℤ≥ ∈ (fBas‘ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∉ wnel 3035 ∀wral 3050 ∃wrex 3059 Vcvv 3461 ∩ cin 3943 ⊆ wss 3944 ∅c0 4322 𝒫 cpw 4604 ran crn 5679 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 1c1 11141 ℤcz 12591 ℤ≥cuz 12855 fBascfbas 21284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-i2m1 11208 ax-1ne0 11209 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-neg 11479 df-nn 12246 df-z 12592 df-uz 12856 df-fbas 21293 |
This theorem is referenced by: uzfbas 23846 |
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