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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 12912 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 ≤ 𝑁) | |
2 | elfzle1 12911 | . 2 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝑁 ≤ 𝐾) | |
3 | elfzelz 12909 | . . 3 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 ∈ ℤ) | |
4 | elfzel2 12907 | . . 3 ⊢ (𝐾 ∈ (𝑁...𝑁) → 𝑁 ∈ ℤ) | |
5 | zre 11986 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
6 | zre 11986 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
7 | letri3 10726 | . . . 4 ⊢ ((𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) | |
8 | 5, 6, 7 | syl2an 597 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 = 𝑁 ↔ (𝐾 ≤ 𝑁 ∧ 𝑁 ≤ 𝐾))) |
9 | 3, 4, 8 | syl2anc 586 | . 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 ≤ cle 10676 ℤcz 11982 ...cfz 12893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-neg 10873 df-z 11983 df-uz 12245 df-fz 12894 |
This theorem is referenced by: fzsn 12950 fz1sbc 12984 fzm1 12988 bccl 13683 hashbc 13812 swrdccatin1 14087 sumsnf 15099 climcnds 15206 prmind2 16029 3prm 16038 vdwlem8 16324 od1 18686 gex1 18716 frgpnabllem1 18993 ply1termlem 24793 coefv0 24838 coemulc 24845 logtayl 25243 leibpilem2 25519 chp1 25744 chtub 25788 2sqlem10 26004 dchrisum0flb 26086 ostth2lem2 26210 axlowdimlem16 26743 sdclem2 35032 0prjspnrel 39289 sumsnd 41303 fourierdlem20 42432 |
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