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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 12883 | . 2 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
2 | eluzfz1 13556 | . 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 2099 ‘cfv 6546 (class class class)co 7416 ℤcz 12604 ℤ≥cuz 12868 ...cfz 13532 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-pre-lttri 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-neg 11488 df-z 12605 df-uz 12869 df-fz 13533 |
This theorem is referenced by: fzsn 13591 fz1sbc 13625 prinfzo0 13719 seqf1o 14057 hashbc 14465 vdwlem8 16985 vdwlem13 16990 coefv0 26272 coemulc 26279 0pthon 30057 metakunt15 41927 metakunt16 41928 |
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