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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 10028 | . . 3 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | 1 | exlimiv 1598 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) |
3 | eluzfz1 10030 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
4 | elex2 2753 | . . 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 126 | 1 ⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∃wex 1492 ∈ wcel 2148 ‘cfv 5216 (class class class)co 5874 ℤ≥cuz 9527 ...cfz 10007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-neg 8130 df-z 9253 df-uz 9528 df-fz 10008 |
This theorem is referenced by: fzn 10041 |
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