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Mirrors > Home > ILE Home > Th. List > elfz1eq | GIF version |
Description: Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.) |
Ref | Expression |
---|---|
elfz1eq | ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzle2 10011 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 ≤ 𝑁) | |
2 | elfzle1 10010 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝑁 ≤ 𝐾) | |
3 | elfzelz 10008 | . . 3 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 ∈ ℤ) | |
4 | elfzel2 10006 | . . 3 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝑁 ∈ ℤ) | |
5 | zre 9243 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
6 | zre 9243 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
7 | letri3 8025 | . . . 4 ⊢ ((𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) | |
8 | 5, 6, 7 | syl2an 289 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) |
9 | 3, 4, 8 | syl2anc 411 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) |
10 | 1, 2, 9 | mpbir2and 944 | 1 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 (class class class)co 5869 ℝcr 7798 ≤ cle 7980 ℤcz 9239 ...cfz 9992 |
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 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-pre-ltirr 7911 ax-pre-apti 7914 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-neg 8118 df-z 9240 df-uz 9515 df-fz 9993 |
This theorem is referenced by: fzsn 10049 fz1sbc 10079 fzm1 10083 fz01or 10094 bccl 10728 sumsnf 11398 prmind2 12100 3prm 12108 2sqlem10 14125 |
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