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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 12535 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
| 3 | 2 | leidd 11829 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ≤ 𝑁) |
| 4 | fznn0 13659 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ (0...𝑁) ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 𝑁))) | |
| 5 | 1, 3, 4 | mpbir2and 713 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0...𝑁)) |
| 6 | elfz3nn0 13661 | . 2 ⊢ (𝑁 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
| 7 | 5, 6 | impbii 209 | 1 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 0cc0 11155 ≤ cle 11296 ℕ0cn0 12526 ...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-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| 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-reu 3381 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-pss 3971 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-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 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-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 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-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 |
| This theorem is referenced by: swrdrlen 14697 pfxid 14722 pfxccat1 14740 pfxpfxid 14747 pfxcctswrd 14748 pfxccatin12 14771 pfxccatid 14779 cshwlen 14837 cshwidxmod 14841 fallfacfac 16081 cayhamlem1 22872 cpmadugsumlemF 22882 wlkepvtx 29678 wlkp1lem7 29697 wlkp1lem8 29698 dfpth2 29749 spthdep 29754 crctcshwlkn0lem6 29835 crctcsh 29844 wwlknllvtx 29866 wwlksnred 29912 wpthswwlks2on 29981 konigsbergiedgw 30267 konigsberglem1 30271 konigsberglem2 30272 konigsberglem3 30273 dlwwlknondlwlknonf1olem1 30383 splfv3 32943 cycpmco2f1 33144 cycpmco2rn 33145 cycpmco2lem3 33148 cycpmco2lem4 33149 cycpmco2lem5 33150 cycpmco2lem6 33151 cycpmco2lem7 33152 cycpmco2 33153 iwrdsplit 34389 fibp1 34403 revpfxsfxrev 35121 poimirlem10 37637 poimirlem17 37644 poimirlem23 37650 poimirlem26 37653 poimirlem27 37654 iccpartiltu 47409 iccpartlt 47411 iccpartleu 47415 iccpartrn 47417 iccelpart 47420 iccpartiun 47421 iccpartdisj 47424 cycl3grtri 47914 usgrexmpl1lem 47980 |
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