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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 11736 | . 2 ⊢ 1 ∈ ℤ | |
2 | zaddcl 11746 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑁 + 1) ∈ ℤ) | |
3 | 1, 2 | mpan2 684 | 1 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 (class class class)co 6906 1c1 10254 + caddc 10256 ℤcz 11705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-n0 11620 df-z 11706 |
This theorem is referenced by: zleltp1 11757 btwnnz 11782 peano2uz2 11794 uzind 11798 uzind2 11799 peano2zd 11814 eluzp1m1 11993 eluzp1p1 11995 peano2uz 12024 zltaddlt1le 12618 elfzp1b 12712 fzval3 12833 fzossfzop1 12842 zesq 13282 hashfzp1 13508 odd2np1lem 15439 odd2np1 15440 mulsucdiv2z 15452 oddp1d2 15457 zob 15458 ltoddhalfle 15460 fldivp1 15973 telgsumfzs 18741 degltp1le 24233 ppiprm 25291 ppinprm 25292 chtprm 25293 chtnprm 25294 chtub 25351 lgsdir2lem2 25465 poimirlem3 33957 poimirlem4 33958 poimirlem16 33970 poimirlem17 33971 poimirlem19 33973 poimirlem20 33974 itg2addnclem2 34006 fdc 34084 ellz1 38175 rmxluc 38345 rmyluc 38346 jm2.27dlem2 38421 fzopredsuc 42222 icceuelpartlem 42260 oddp1evenALTV 42418 elfzolborelfzop1 43157 dignn0flhalflem1 43257 |
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