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Mirrors > Home > MPE Home > Th. List > Mathboxes > allbutfi | Structured version Visualization version GIF version |
Description: For all but finitely many. Some authors say "cofinitely many". Some authors say "ultimately". Compare with eliuniin 44530 and eliuniin2 44551 (here, the precondition can be dropped; see eliuniincex 44540). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
allbutfi.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
allbutfi.a | ⊢ 𝐴 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 |
Ref | Expression |
---|---|
allbutfi | ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | allbutfi.a | . . . . . 6 ⊢ 𝐴 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 | |
2 | 1 | eleq2i 2817 | . . . . 5 ⊢ (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵) |
3 | 2 | biimpi 215 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵) |
4 | eliun 4995 | . . . 4 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵) | |
5 | 3, 4 | sylib 217 | . . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵) |
6 | nfcv 2892 | . . . . 5 ⊢ Ⅎ𝑛𝑋 | |
7 | nfiu1 5025 | . . . . . 6 ⊢ Ⅎ𝑛∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 | |
8 | 1, 7 | nfcxfr 2890 | . . . . 5 ⊢ Ⅎ𝑛𝐴 |
9 | 6, 8 | nfel 2907 | . . . 4 ⊢ Ⅎ𝑛 𝑋 ∈ 𝐴 |
10 | eliin 4996 | . . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)) | |
11 | 10 | biimpd 228 | . . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 → ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)) |
12 | 11 | a1d 25 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑛 ∈ 𝑍 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 → ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵))) |
13 | 9, 12 | reximdai 3249 | . . 3 ⊢ (𝑋 ∈ 𝐴 → (∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)) |
14 | 5, 13 | mpd 15 | . 2 ⊢ (𝑋 ∈ 𝐴 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) |
15 | simpr 483 | . . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) → ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) | |
16 | allbutfi.z | . . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
17 | 16 | eleq2i 2817 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀)) |
18 | 17 | biimpi 215 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀)) |
19 | eluzelz 12862 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ) | |
20 | uzid 12867 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛)) | |
21 | 18, 19, 20 | 3syl 18 | . . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑛)) |
22 | 21 | ne0d 4331 | . . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅) |
23 | eliin2 44547 | . . . . . . . . 9 ⊢ ((ℤ≥‘𝑛) ≠ ∅ → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)) | |
24 | 22, 23 | syl 17 | . . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)) |
25 | 24 | adantr 479 | . . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵)) |
26 | 15, 25 | mpbird 256 | . . . . . 6 ⊢ ((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵) |
27 | 26 | ex 411 | . . . . 5 ⊢ (𝑛 ∈ 𝑍 → (∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵)) |
28 | 27 | reximia 3071 | . . . 4 ⊢ (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵) |
29 | 28, 4 | sylibr 233 | . . 3 ⊢ (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵) |
30 | 29, 1 | eleqtrrdi 2836 | . 2 ⊢ (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐴) |
31 | 14, 30 | impbii 208 | 1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 ∅c0 4318 ∪ ciun 4991 ∩ ciin 4992 ‘cfv 6543 ℤcz 12588 ℤ≥cuz 12852 |
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 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-pre-lttri 11212 |
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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-neg 11477 df-z 12589 df-uz 12853 |
This theorem is referenced by: allbutfiinf 44865 allbutfifvre 45126 smflimlem3 46224 smfliminflem 46281 |
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