![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elfz1eq | Structured version Visualization version 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 12906 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 ≤ 𝑁) | |
2 | elfzle1 12905 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝑁 ≤ 𝐾) | |
3 | elfzelz 12902 | . . 3 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 ∈ ℤ) | |
4 | elfzel2 12900 | . . 3 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝑁 ∈ ℤ) | |
5 | zre 11973 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
6 | zre 11973 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
7 | letri3 10715 | . . . 4 ⊢ ((𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) | |
8 | 5, 6, 7 | syl2an 598 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) |
9 | 3, 4, 8 | syl2anc 587 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) |
10 | 1, 2, 9 | mpbir2and 712 | 1 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 ≤ cle 10665 ℤcz 11969 ...cfz 12885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-z 11970 df-uz 12232 df-fz 12886 |
This theorem is referenced by: fzsn 12944 fz1sbc 12978 fzm1 12982 bccl 13678 hashbc 13807 swrdccatin1 14078 sumsnf 15091 climcnds 15198 prmind2 16019 3prm 16028 vdwlem8 16314 od1 18678 gex1 18708 frgpnabllem1 18986 ply1termlem 24800 coefv0 24845 coemulc 24852 logtayl 25251 leibpilem2 25527 chp1 25752 chtub 25796 2sqlem10 26012 dchrisum0flb 26094 ostth2lem2 26218 axlowdimlem16 26751 sdclem2 35180 0prjspnrel 39613 sumsnd 41655 fourierdlem20 42769 |
Copyright terms: Public domain | W3C validator |