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Mirrors > Home > MPE Home > Th. List > peano2nn | Structured version Visualization version GIF version |
Description: Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
peano2nn | ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frfnom 8061 | . . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) Fn ω | |
2 | fvelrnb 6720 | . . . 4 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) Fn ω → (𝐴 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) = 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (𝐴 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) = 𝐴) |
4 | ovex 7178 | . . . . . . 7 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1) ∈ V | |
5 | eqid 2821 | . . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) | |
6 | oveq1 7152 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 + 1) = (𝑥 + 1)) | |
7 | oveq1 7152 | . . . . . . . 8 ⊢ (𝑧 = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) → (𝑧 + 1) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1)) | |
8 | 5, 6, 7 | frsucmpt2 8067 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1) ∈ V) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1)) |
9 | 4, 8 | mpan2 687 | . . . . . 6 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1)) |
10 | peano2 7590 | . . . . . . . 8 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω) | |
11 | fnfvelrn 6841 | . . . . . . . 8 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) Fn ω ∧ suc 𝑦 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘suc 𝑦) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)) | |
12 | 1, 10, 11 | sylancr 587 | . . . . . . 7 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘suc 𝑦) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)) |
13 | df-nn 11628 | . . . . . . . 8 ⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) | |
14 | df-ima 5562 | . . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) | |
15 | 13, 14 | eqtri 2844 | . . . . . . 7 ⊢ ℕ = ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) |
16 | 12, 15 | eleqtrrdi 2924 | . . . . . 6 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘suc 𝑦) ∈ ℕ) |
17 | 9, 16 | eqeltrrd 2914 | . . . . 5 ⊢ (𝑦 ∈ ω → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1) ∈ ℕ) |
18 | oveq1 7152 | . . . . . 6 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) = 𝐴 → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1) = (𝐴 + 1)) | |
19 | 18 | eleq1d 2897 | . . . . 5 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) = 𝐴 → ((((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1) ∈ ℕ ↔ (𝐴 + 1) ∈ ℕ)) |
20 | 17, 19 | syl5ibcom 246 | . . . 4 ⊢ (𝑦 ∈ ω → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) = 𝐴 → (𝐴 + 1) ∈ ℕ)) |
21 | 20 | rexlimiv 3280 | . . 3 ⊢ (∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) = 𝐴 → (𝐴 + 1) ∈ ℕ) |
22 | 3, 21 | sylbi 218 | . 2 ⊢ (𝐴 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) → (𝐴 + 1) ∈ ℕ) |
23 | 22, 15 | eleq2s 2931 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∃wrex 3139 Vcvv 3495 ↦ cmpt 5138 ran crn 5550 ↾ cres 5551 “ cima 5552 suc csuc 6187 Fn wfn 6344 ‘cfv 6349 (class class class)co 7145 ωcom 7568 reccrdg 8036 1c1 10527 + caddc 10529 ℕcn 11627 |
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 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
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 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-nn 11628 |
This theorem is referenced by: dfnn2 11640 dfnn3 11641 peano2nnd 11644 nnind 11645 nnaddcl 11649 2nn 11699 3nn 11705 4nn 11709 5nn 11712 6nn 11715 7nn 11718 8nn 11721 9nn 11724 nnunb 11882 nneo 12055 10nn 12103 fzonn0p1p1 13106 ser1const 13416 expp1 13426 facp1 13628 relexpsucnnl 14381 isercolllem1 15011 isercoll2 15015 climcndslem2 15195 climcnds 15196 harmonic 15204 trireciplem 15207 trirecip 15208 rpnnen2lem9 15565 sqrt2irr 15592 nno 15723 nnoddm1d2 15727 rplpwr 15897 prmind2 16019 eulerthlem2 16109 pcmpt 16218 pockthi 16233 prmreclem6 16247 dec5nprm 16392 mulgnnp1 18176 chfacfisf 21392 chfacfisfcpmat 21393 cayhamlem1 21404 1stcfb 21983 bcthlem3 23858 bcthlem4 23859 ovolunlem1a 24026 ovolicc2lem4 24050 voliunlem1 24080 volsup 24086 volsup2 24135 itg1climres 24244 mbfi1fseqlem5 24249 itg2monolem1 24280 itg2i1fseqle 24284 itg2i1fseq 24285 itg2i1fseq2 24286 itg2addlem 24288 itg2gt0 24290 itg2cnlem1 24291 aaliou3lem7 24867 emcllem1 25501 emcllem2 25502 emcllem3 25503 emcllem5 25505 emcllem6 25506 emcllem7 25507 zetacvg 25520 lgam1 25569 bclbnd 25784 bposlem5 25792 2sqlem10 25932 dchrisumlem2 25994 logdivbnd 26060 pntrsumo1 26069 pntrsumbnd 26070 wwlksext2clwwlk 27764 numclwwlk2lem1 28083 numclwlk2lem2f 28084 opsqrlem5 29849 opsqrlem6 29850 nnindf 30462 psgnfzto1st 30675 esumpmono 31238 fibp1 31559 rrvsum 31612 subfacp1lem6 32330 subfaclim 32333 bcprod 32868 bccolsum 32869 iprodgam 32872 faclimlem1 32873 faclimlem2 32874 faclim2 32878 nn0prpwlem 33568 mblfinlem2 34812 volsupnfl 34819 seqpo 34905 incsequz 34906 incsequz2 34907 geomcau 34917 heiborlem6 34977 bfplem1 34983 jm2.27dlem4 39489 nnsplit 41506 sumnnodd 41791 stoweidlem20 42186 wallispilem4 42234 wallispi2lem1 42237 wallispi2lem2 42238 stirlinglem4 42243 stirlinglem8 42247 stirlinglem11 42250 stirlinglem12 42251 stirlinglem13 42252 vonioolem2 42844 vonicclem2 42847 deccarry 43392 iccpartres 43425 iccelpart 43440 odz2prm2pw 43572 fmtnoprmfac1 43574 fmtnoprmfac2 43576 lighneallem4 43622 |
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