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Mirrors > Home > MPE Home > Th. List > peano2uz | Structured version Visualization version GIF version |
Description: Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.) |
Ref | Expression |
---|---|
peano2uz | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1128 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℤ) | |
2 | peano2z 12011 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
3 | 2 | 3ad2ant2 1126 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑁 + 1) ∈ ℤ) |
4 | zre 11973 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
5 | zre 11973 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | letrp1 11472 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) | |
7 | 5, 6 | syl3an2 1156 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) |
8 | 4, 7 | syl3an1 1155 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ (𝑁 + 1)) |
9 | 1, 3, 8 | 3jca 1120 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑀 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑁 + 1))) |
10 | eluz2 12237 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
11 | eluz2 12237 | . 2 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑁 + 1))) | |
12 | 9, 10, 11 | 3imtr4i 293 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 1c1 10526 + caddc 10528 ≤ cle 10664 ℤcz 11969 ℤ≥cuz 12231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 |
This theorem is referenced by: peano2uzs 12290 peano2uzr 12291 uzaddcl 12292 fzsplit 12921 fzssp1 12938 fzsuc 12942 fzpred 12943 fzp1ss 12946 fzp1elp1 12948 fztp 12951 fzneuz 12976 fzosplitsnm1 13100 fzofzp1 13122 fzosplitsn 13133 fzosplitpr 13134 fzostep1 13141 om2uzuzi 13305 uzrdgsuci 13316 fzen2 13325 fzfi 13328 seqsplit 13391 seqf1olem1 13397 seqf1olem2 13398 seqz 13406 faclbnd3 13640 bcm1k 13663 seqcoll 13810 seqcoll2 13811 swrds1 14016 pfxccatpfx2 14087 clim2ser 14999 clim2ser2 15000 serf0 15025 iseraltlem2 15027 iseralt 15029 fsump1 15099 fsump1i 15112 fsumparts 15149 cvgcmp 15159 isum1p 15184 isumsup2 15189 climcndslem1 15192 climcndslem2 15193 climcnds 15194 cvgrat 15227 mertenslem1 15228 clim2prod 15232 clim2div 15233 ntrivcvgfvn0 15243 fprodntriv 15284 fprodp1 15311 fprodabs 15316 binomfallfaclem2 15382 pcfac 16223 gsumsplit1r 17885 gsumprval 17886 telgsumfzslem 19037 telgsumfzs 19038 dvply2g 24801 aaliou3lem2 24859 ppinprm 25656 chtnprm 25658 ppiublem1 25705 chtublem 25714 chtub 25715 bposlem6 25792 pntlemf 26108 ostth2lem2 26137 clwwlkvbij 27819 fzsplit3 30443 esumcvg 31244 sseqf 31549 gsumnunsn 31710 signstfvp 31740 iprodefisumlem 32869 poimirlem1 34774 poimirlem2 34775 poimirlem3 34776 poimirlem4 34777 poimirlem6 34779 poimirlem7 34780 poimirlem8 34781 poimirlem9 34782 poimirlem12 34785 poimirlem13 34786 poimirlem14 34787 poimirlem15 34788 poimirlem16 34789 poimirlem17 34790 poimirlem18 34791 poimirlem19 34792 poimirlem20 34793 poimirlem21 34794 poimirlem22 34795 poimirlem23 34796 poimirlem24 34797 poimirlem26 34799 poimirlem27 34800 poimirlem31 34804 poimirlem32 34805 sdclem2 34898 fdc 34901 mettrifi 34913 bfplem2 34982 rexrabdioph 39269 monotuz 39416 wallispilem1 42227 dirkertrigeqlem2 42261 sge0p1 42573 carageniuncllem1 42680 iccpartres 43455 iccelpart 43470 fmtno4prm 43614 |
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