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Mirrors > Home > MPE Home > Th. List > nn0fz0 | Structured version Visualization version GIF version |
Description: A nonnegative integer is always part of the finite set of sequential nonnegative integers with this integer as upper bound. (Contributed by Scott Fenton, 21-Mar-2018.) |
Ref | Expression |
---|---|
nn0fz0 | ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
2 | nn0re 12503 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
3 | 2 | leidd 11802 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ≤ 𝑁) |
4 | fznn0 13617 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ (0...𝑁) ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 𝑁))) | |
5 | 1, 3, 4 | mpbir2and 712 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0...𝑁)) |
6 | elfz3nn0 13619 | . 2 ⊢ (𝑁 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
7 | 5, 6 | impbii 208 | 1 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2099 class class class wbr 5142 (class class class)co 7414 0cc0 11130 ≤ cle 11271 ℕ0cn0 12494 ...cfz 13508 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 |
This theorem is referenced by: swrdrlen 14633 pfxid 14658 pfxccat1 14676 pfxpfxid 14683 pfxcctswrd 14684 pfxccatin12 14707 pfxccatid 14715 cshwlen 14773 cshwidxmod 14777 fallfacfac 16013 cayhamlem1 22755 cpmadugsumlemF 22765 wlkepvtx 29461 wlkp1lem7 29480 wlkp1lem8 29481 spthdep 29535 crctcshwlkn0lem6 29613 crctcsh 29622 wwlknllvtx 29644 wwlksnred 29690 wpthswwlks2on 29759 konigsbergiedgw 30045 konigsberglem1 30049 konigsberglem2 30050 konigsberglem3 30051 dlwwlknondlwlknonf1olem1 30161 splfv3 32661 cycpmco2f1 32823 cycpmco2rn 32824 cycpmco2lem3 32827 cycpmco2lem4 32828 cycpmco2lem5 32829 cycpmco2lem6 32830 cycpmco2lem7 32831 cycpmco2 32832 iwrdsplit 33943 fibp1 33957 revpfxsfxrev 34661 poimirlem10 37038 poimirlem17 37045 poimirlem23 37051 poimirlem26 37054 poimirlem27 37055 iccpartiltu 46685 iccpartlt 46687 iccpartleu 46691 iccpartrn 46693 iccelpart 46696 iccpartiun 46697 iccpartdisj 46700 |
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