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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 9802 | . . 3 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | 1 | exlimiv 1577 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) |
3 | eluzfz1 9804 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
4 | elex2 2697 | . . 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 125 | 1 ⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∃wex 1468 ∈ wcel 1480 ‘cfv 5118 (class class class)co 5767 ℤ≥cuz 9319 ...cfz 9783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-neg 7929 df-z 9048 df-uz 9320 df-fz 9784 |
This theorem is referenced by: fzn 9815 |
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