![]() |
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 13509 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 ≤ 𝑁) | |
2 | elfzle1 13508 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝑁 ≤ 𝐾) | |
3 | elfzelz 13505 | . . 3 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 ∈ ℤ) | |
4 | elfzel2 13503 | . . 3 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝑁 ∈ ℤ) | |
5 | zre 12566 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
6 | zre 12566 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
7 | letri3 11303 | . . . 4 ⊢ ((𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) | |
8 | 5, 6, 7 | syl2an 596 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) |
9 | 3, 4, 8 | syl2anc 584 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) |
10 | 1, 2, 9 | mpbir2and 711 | 1 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7411 ℝcr 11111 ≤ cle 11253 ℤcz 12562 ...cfz 13488 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-neg 11451 df-z 12563 df-uz 12827 df-fz 13489 |
This theorem is referenced by: fzsn 13547 fz1sbc 13581 fzm1 13585 bccl 14286 hashbc 14416 swrdccatin1 14679 sumsnf 15693 climcnds 15801 prmind2 16626 3prm 16635 vdwlem8 16925 od1 19468 gex1 19500 frgpnabllem1 19782 ply1termlem 25941 coefv0 25986 coemulc 25993 logtayl 26392 leibpilem2 26670 chp1 26895 chtub 26939 2sqlem10 27155 dchrisum0flb 27237 ostth2lem2 27361 axlowdimlem16 28470 sdclem2 36913 0prjspnrel 41671 sumsnd 44012 fourierdlem20 45142 |
Copyright terms: Public domain | W3C validator |