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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 12359 | . 2 ⊢ 1 ∈ ℤ | |
2 | zaddcl 12369 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑁 + 1) ∈ ℤ) | |
3 | 1, 2 | mpan2 688 | 1 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 (class class class)co 7284 1c1 10881 + caddc 10883 ℤcz 12328 |
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 2710 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-om 7722 df-2nd 7841 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-er 8507 df-en 8743 df-dom 8744 df-sdom 8745 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-nn 11983 df-n0 12243 df-z 12329 |
This theorem is referenced by: zleltp1 12380 btwnnz 12405 peano2uz2 12417 uzind 12421 uzind2 12422 peano2zd 12438 eluzp1m1 12617 eluzp1p1 12619 peano2uz 12650 zltaddlt1le 13246 elfzp1b 13342 fzval3 13465 fzossfzop1 13474 zesq 13950 hashfzp1 14155 odd2np1lem 16058 odd2np1 16059 mulsucdiv2z 16071 oddp1d2 16076 zob 16077 ltoddhalfle 16079 fldivp1 16607 telgsumfzs 19599 degltp1le 25247 ppiprm 26309 ppinprm 26310 chtprm 26311 chtnprm 26312 chtub 26369 lgsdir2lem2 26483 poimirlem3 35789 poimirlem4 35790 poimirlem16 35802 poimirlem17 35803 poimirlem19 35805 poimirlem20 35806 itg2addnclem2 35838 fdc 35912 ellz1 40596 rmxluc 40765 rmyluc 40766 jm2.27dlem2 40839 fzopredsuc 44826 icceuelpartlem 44898 oddp1evenALTV 45139 elfzolborelfzop1 45871 dignn0flhalflem1 45972 |
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