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Mirrors > Home > MPE Home > Th. List > peano2z | Structured version Visualization version GIF version |
Description: Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.) |
Ref | Expression |
---|---|
peano2z | ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 12106 | . 2 ⊢ 1 ∈ ℤ | |
2 | zaddcl 12116 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑁 + 1) ∈ ℤ) | |
3 | 1, 2 | mpan2 691 | 1 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 (class class class)co 7183 1c1 10629 + caddc 10631 ℤcz 12075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-n0 11990 df-z 12076 |
This theorem is referenced by: zleltp1 12127 btwnnz 12152 peano2uz2 12164 uzind 12168 uzind2 12169 peano2zd 12184 eluzp1m1 12363 eluzp1p1 12365 peano2uz 12396 zltaddlt1le 12992 elfzp1b 13088 fzval3 13210 fzossfzop1 13219 zesq 13692 hashfzp1 13897 odd2np1lem 15798 odd2np1 15799 mulsucdiv2z 15811 oddp1d2 15816 zob 15817 ltoddhalfle 15819 fldivp1 16346 telgsumfzs 19241 degltp1le 24839 ppiprm 25901 ppinprm 25902 chtprm 25903 chtnprm 25904 chtub 25961 lgsdir2lem2 26075 poimirlem3 35436 poimirlem4 35437 poimirlem16 35449 poimirlem17 35450 poimirlem19 35452 poimirlem20 35453 itg2addnclem2 35485 fdc 35559 ellz1 40202 rmxluc 40371 rmyluc 40372 jm2.27dlem2 40445 fzopredsuc 44397 icceuelpartlem 44469 oddp1evenALTV 44710 elfzolborelfzop1 45442 dignn0flhalflem1 45543 |
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