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| Mirrors > Home > ILE Home > Th. List > elfznn0 | GIF version | ||
| Description: A member of a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
| Ref | Expression |
|---|---|
| elfznn0 | ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfz2nn0 10234 | . 2 ⊢ (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁)) | |
| 2 | 1 | simp1bi 1015 | 1 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∈ wcel 2176 class class class wbr 4044 (class class class)co 5944 0cc0 7925 ≤ cle 8108 ℕ0cn0 9295 ...cfz 10130 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-addcom 8025 ax-addass 8027 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-0id 8033 ax-rnegex 8034 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-ltadd 8041 |
| This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-inn 9037 df-n0 9296 df-z 9373 df-uz 9649 df-fz 10131 |
| This theorem is referenced by: fz0ssnn0 10238 fz0fzdiffz0 10252 difelfzle 10256 fzo0ssnn0 10344 bcval 10894 bcrpcl 10898 bccmpl 10899 bcp1n 10906 bcp1nk 10907 permnn 10916 binomlem 11794 binom1p 11796 binom1dif 11798 bcxmas 11800 arisum 11809 arisum2 11810 pwm1geoserap1 11819 geo2sum 11825 mertenslemub 11845 mertenslemi1 11846 mertenslem2 11847 mertensabs 11848 efcvgfsum 11978 efaddlem 11985 eirraplem 12088 3dvds 12175 bitsfzolem 12265 prmdiveq 12558 hashgcdlem 12560 pcbc 12674 ennnfonelemim 12795 ctinfomlemom 12798 elply2 15207 plyf 15209 elplyd 15213 ply1termlem 15214 plyaddlem1 15219 plymullem1 15220 plyaddlem 15221 plymullem 15222 plycoeid3 15229 plycolemc 15230 plycjlemc 15232 plycj 15233 plycn 15234 plyrecj 15235 dvply1 15237 dvply2g 15238 dvdsppwf1o 15461 sgmppw 15464 1sgmprm 15466 mersenne 15469 lgseisenlem1 15547 |
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