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Mirrors > Home > ILE Home > Th. List > elfzp12 | GIF version |
Description: Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.) |
Ref | Expression |
---|---|
elfzp12 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzelz 9799 | . . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ) | |
2 | 1 | anim2i 339 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ)) |
3 | eluzel2 9324 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
4 | eleq1 2200 | . . . . 5 ⊢ (𝐾 = 𝑀 → (𝐾 ∈ ℤ ↔ 𝑀 ∈ ℤ)) | |
5 | 3, 4 | syl5ibrcom 156 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 = 𝑀 → 𝐾 ∈ ℤ)) |
6 | 5 | imdistani 441 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 = 𝑀) → (𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ)) |
7 | elfzelz 9799 | . . . 4 ⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → 𝐾 ∈ ℤ) | |
8 | 7 | anim2i 339 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ)) |
9 | 6, 8 | jaodan 786 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))) → (𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ)) |
10 | fzpred 9843 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) | |
11 | 10 | eleq2d 2207 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ∈ ({𝑀} ∪ ((𝑀 + 1)...𝑁)))) |
12 | elun 3212 | . . . 4 ⊢ (𝐾 ∈ ({𝑀} ∪ ((𝑀 + 1)...𝑁)) ↔ (𝐾 ∈ {𝑀} ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))) | |
13 | 11, 12 | syl6bb 195 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ {𝑀} ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))) |
14 | elsng 3537 | . . . 4 ⊢ (𝐾 ∈ ℤ → (𝐾 ∈ {𝑀} ↔ 𝐾 = 𝑀)) | |
15 | 14 | orbi1d 780 | . . 3 ⊢ (𝐾 ∈ ℤ → ((𝐾 ∈ {𝑀} ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))) |
16 | 13, 15 | sylan9bb 457 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))) |
17 | 2, 9, 16 | pm5.21nd 901 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ∪ cun 3064 {csn 3522 ‘cfv 5118 (class class class)co 5767 1c1 7614 + caddc 7616 ℤcz 9047 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-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 |
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-reu 2421 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-int 3767 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-riota 5723 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-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 |
This theorem is referenced by: bcpasc 10505 |
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