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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 9801 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 ≤ 𝑁) | |
2 | elfzle1 9800 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝑁 ≤ 𝐾) | |
3 | elfzelz 9799 | . . 3 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 ∈ ℤ) | |
4 | elfzel2 9797 | . . 3 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝑁 ∈ ℤ) | |
5 | zre 9051 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
6 | zre 9051 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
7 | letri3 7838 | . . . 4 ⊢ ((𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) | |
8 | 5, 6, 7 | syl2an 287 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) |
9 | 3, 4, 8 | syl2anc 408 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) |
10 | 1, 2, 9 | mpbir2and 928 | 1 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℝcr 7612 ≤ cle 7794 ℤcz 9047 ...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-apti 7728 |
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: fzsn 9839 fz1sbc 9869 fzm1 9873 fz01or 9884 bccl 10506 sumsnf 11171 prmind2 11790 3prm 11798 |
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