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| Mirrors > Home > MPE Home > Th. List > elfz3 | Structured version Visualization version GIF version | ||
| Description: Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.) | 
| Ref | Expression | 
|---|---|
| elfz3 | ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uzid 12893 | . 2 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
| 2 | eluzfz1 13571 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑁) → 𝑁 ∈ (𝑁...𝑁)) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ℤcz 12613 ℤ≥cuz 12878 ...cfz 13547 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 df-fz 13548 | 
| This theorem is referenced by: fzsn 13606 fz1sbc 13640 prinfzo0 13738 seqf1o 14084 hashbc 14492 vdwlem8 17026 vdwlem13 17031 coefv0 26287 coemulc 26294 0pthon 30146 metakunt15 42220 metakunt16 42221 | 
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