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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 9048 | . . 3 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | 1 | exlimiv 1529 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) |
3 | eluzfz1 9050 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
4 | elex2 2615 | . . 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 124 | 1 ⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∃wex 1421 ∈ wcel 1433 ‘cfv 4922 (class class class)co 5532 ℤ≥cuz 8619 ...cfz 9029 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-neg 7282 df-z 8352 df-uz 8620 df-fz 9030 |
This theorem is referenced by: fzn 9061 |
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