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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 12250 | . 2 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
2 | eluzfz1 12906 | . 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 6348 (class class class)co 7148 ℤcz 11973 ℤ≥cuz 12235 ...cfz 12884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-pre-lttri 10603 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-neg 10865 df-z 11974 df-uz 12236 df-fz 12885 |
This theorem is referenced by: fzsn 12941 fz1sbc 12975 prinfzo0 13068 seqf1o 13403 hashbc 13803 vdwlem8 16316 vdwlem13 16321 coefv0 24830 coemulc 24837 0pthon 27898 |
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