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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 12423 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
3 | 2 | leidd 11722 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ≤ 𝑁) |
4 | fznn0 13534 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ (0...𝑁) ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 𝑁))) | |
5 | 1, 3, 4 | mpbir2and 712 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0...𝑁)) |
6 | elfz3nn0 13536 | . 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 2107 class class class wbr 5106 (class class class)co 7358 0cc0 11052 ≤ cle 11191 ℕ0cn0 12414 ...cfz 13425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 |
This theorem is referenced by: swrdrlen 14548 pfxid 14573 pfxccat1 14591 pfxpfxid 14598 pfxcctswrd 14599 pfxccatin12 14622 pfxccatid 14630 cshwlen 14688 cshwidxmod 14692 fallfacfac 15929 cayhamlem1 22218 cpmadugsumlemF 22228 wlkepvtx 28611 wlkp1lem7 28630 wlkp1lem8 28631 spthdep 28685 crctcshwlkn0lem6 28763 crctcsh 28772 wwlknllvtx 28794 wwlksnred 28840 wpthswwlks2on 28909 konigsbergiedgw 29195 konigsberglem1 29199 konigsberglem2 29200 konigsberglem3 29201 dlwwlknondlwlknonf1olem1 29311 splfv3 31815 cycpmco2f1 31976 cycpmco2rn 31977 cycpmco2lem3 31980 cycpmco2lem4 31981 cycpmco2lem5 31982 cycpmco2lem6 31983 cycpmco2lem7 31984 cycpmco2 31985 iwrdsplit 32990 fibp1 33004 revpfxsfxrev 33712 poimirlem10 36091 poimirlem17 36098 poimirlem23 36104 poimirlem26 36107 poimirlem27 36108 iccpartiltu 45621 iccpartlt 45623 iccpartleu 45627 iccpartrn 45629 iccelpart 45632 iccpartiun 45633 iccpartdisj 45636 |
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