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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 11890 | . . 3 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
2 | frn 6193 | . . 3 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ran ℤ≥ ⊆ 𝒫 ℤ) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ran ℤ≥ ⊆ 𝒫 ℤ |
4 | ffn 6185 | . . . . . 6 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ ℤ≥ Fn ℤ |
6 | 1z 11608 | . . . . 5 ⊢ 1 ∈ ℤ | |
7 | fnfvelrn 6499 | . . . . 5 ⊢ ((ℤ≥ Fn ℤ ∧ 1 ∈ ℤ) → (ℤ≥‘1) ∈ ran ℤ≥) | |
8 | 5, 6, 7 | mp2an 664 | . . . 4 ⊢ (ℤ≥‘1) ∈ ran ℤ≥ |
9 | 8 | ne0ii 4071 | . . 3 ⊢ ran ℤ≥ ≠ ∅ |
10 | uzid 11902 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ (ℤ≥‘𝑥)) | |
11 | n0i 4068 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘𝑥) → ¬ (ℤ≥‘𝑥) = ∅) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → ¬ (ℤ≥‘𝑥) = ∅) |
13 | 12 | nrex 3148 | . . . . 5 ⊢ ¬ ∃𝑥 ∈ ℤ (ℤ≥‘𝑥) = ∅ |
14 | fvelrnb 6385 | . . . . . 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 312 | . . . 4 ⊢ ¬ ∅ ∈ ran ℤ≥ |
17 | 16 | nelir 3049 | . . 3 ⊢ ∅ ∉ ran ℤ≥ |
18 | uzin2 14291 | . . . . 5 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) → (𝑥 ∩ 𝑦) ∈ ran ℤ≥) | |
19 | vex 3354 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
20 | 19 | inex1 4933 | . . . . . 6 ⊢ (𝑥 ∩ 𝑦) ∈ V |
21 | 20 | pwid 4313 | . . . . 5 ⊢ (𝑥 ∩ 𝑦) ∈ 𝒫 (𝑥 ∩ 𝑦) |
22 | inelcm 4175 | . . . . 5 ⊢ (((𝑥 ∩ 𝑦) ∈ ran ℤ≥ ∧ (𝑥 ∩ 𝑦) ∈ 𝒫 (𝑥 ∩ 𝑦)) → (ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅) | |
23 | 18, 21, 22 | sylancl 566 | . . . 4 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) → (ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅) |
24 | 23 | rgen2a 3126 | . . 3 ⊢ ∀𝑥 ∈ ran ℤ≥∀𝑦 ∈ ran ℤ≥(ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ |
25 | 9, 17, 24 | 3pm3.2i 1423 | . 2 ⊢ (ran ℤ≥ ≠ ∅ ∧ ∅ ∉ ran ℤ≥ ∧ ∀𝑥 ∈ ran ℤ≥∀𝑦 ∈ ran ℤ≥(ran ℤ≥ ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅) |
26 | zex 11587 | . . 3 ⊢ ℤ ∈ V | |
27 | isfbas 21852 | . . 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 682 | 1 ⊢ ran ℤ≥ ∈ (fBas‘ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∉ wnel 3046 ∀wral 3061 ∃wrex 3062 Vcvv 3351 ∩ cin 3722 ⊆ wss 3723 ∅c0 4063 𝒫 cpw 4297 ran crn 5250 Fn wfn 6026 ⟶wf 6027 ‘cfv 6031 1c1 10138 ℤcz 11578 ℤ≥cuz 11887 fBascfbas 19948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-z 11579 df-uz 11888 df-fbas 19957 |
This theorem is referenced by: uzfbas 21921 |
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