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Mirrors > Home > ILE Home > Th. List > fzm | GIF version |
Description: Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.) |
Ref | Expression |
---|---|
fzm | ⊢ (∃x x ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz2 8663 | . . 3 ⊢ (x ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | 1 | exlimiv 1486 | . 2 ⊢ (∃x x ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) |
3 | eluzfz1 8665 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
4 | elex2 2564 | . . 3 ⊢ (𝑀 ∈ (𝑀...𝑁) → ∃x x ∈ (𝑀...𝑁)) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ∃x x ∈ (𝑀...𝑁)) |
6 | 2, 5 | impbii 117 | 1 ⊢ (∃x x ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∃wex 1378 ∈ wcel 1390 ‘cfv 4845 (class class class)co 5455 ℤ≥cuz 8249 ...cfz 8644 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-pre-ltirr 6795 ax-pre-ltwlin 6796 |
This theorem depends on definitions: df-bi 110 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-neg 6982 df-z 8022 df-uz 8250 df-fz 8645 |
This theorem is referenced by: fzn 8676 |
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