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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 12478 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
3 | 2 | leidd 11777 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ≤ 𝑁) |
4 | fznn0 13590 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ (0...𝑁) ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 𝑁))) | |
5 | 1, 3, 4 | mpbir2and 710 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0...𝑁)) |
6 | elfz3nn0 13592 | . 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 2098 class class class wbr 5138 (class class class)co 7401 0cc0 11106 ≤ cle 11246 ℕ0cn0 12469 ...cfz 13481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 |
This theorem is referenced by: swrdrlen 14606 pfxid 14631 pfxccat1 14649 pfxpfxid 14656 pfxcctswrd 14657 pfxccatin12 14680 pfxccatid 14688 cshwlen 14746 cshwidxmod 14750 fallfacfac 15986 cayhamlem1 22690 cpmadugsumlemF 22700 wlkepvtx 29386 wlkp1lem7 29405 wlkp1lem8 29406 spthdep 29460 crctcshwlkn0lem6 29538 crctcsh 29547 wwlknllvtx 29569 wwlksnred 29615 wpthswwlks2on 29684 konigsbergiedgw 29970 konigsberglem1 29974 konigsberglem2 29975 konigsberglem3 29976 dlwwlknondlwlknonf1olem1 30086 splfv3 32589 cycpmco2f1 32751 cycpmco2rn 32752 cycpmco2lem3 32755 cycpmco2lem4 32756 cycpmco2lem5 32757 cycpmco2lem6 32758 cycpmco2lem7 32759 cycpmco2 32760 iwrdsplit 33875 fibp1 33889 revpfxsfxrev 34595 poimirlem10 36988 poimirlem17 36995 poimirlem23 37001 poimirlem26 37004 poimirlem27 37005 iccpartiltu 46575 iccpartlt 46577 iccpartleu 46581 iccpartrn 46583 iccelpart 46586 iccpartiun 46587 iccpartdisj 46590 |
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