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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 8072 | . . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) Fn ω | |
2 | fvelrnb 6728 | . . . 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 7191 | . . . . . . 7 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1) ∈ V | |
5 | eqid 2823 | . . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) | |
6 | oveq1 7165 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 + 1) = (𝑥 + 1)) | |
7 | oveq1 7165 | . . . . . . . 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 8078 | . . . . . . 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 689 | . . . . . 6 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1)) |
10 | peano2 7604 | . . . . . . . 8 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω) | |
11 | fnfvelrn 6850 | . . . . . . . 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 589 | . . . . . . 7 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘suc 𝑦) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)) |
13 | df-nn 11641 | . . . . . . . 8 ⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) | |
14 | df-ima 5570 | . . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) | |
15 | 13, 14 | eqtri 2846 | . . . . . . 7 ⊢ ℕ = ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) |
16 | 12, 15 | eleqtrrdi 2926 | . . . . . 6 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘suc 𝑦) ∈ ℕ) |
17 | 9, 16 | eqeltrrd 2916 | . . . . 5 ⊢ (𝑦 ∈ ω → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1) ∈ ℕ) |
18 | oveq1 7165 | . . . . . 6 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) = 𝐴 → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1) = (𝐴 + 1)) | |
19 | 18 | eleq1d 2899 | . . . . 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 247 | . . . 4 ⊢ (𝑦 ∈ ω → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) = 𝐴 → (𝐴 + 1) ∈ ℕ)) |
21 | 20 | rexlimiv 3282 | . . 3 ⊢ (∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) = 𝐴 → (𝐴 + 1) ∈ ℕ) |
22 | 3, 21 | sylbi 219 | . 2 ⊢ (𝐴 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) → (𝐴 + 1) ∈ ℕ) |
23 | 22, 15 | eleq2s 2933 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 Vcvv 3496 ↦ cmpt 5148 ran crn 5558 ↾ cres 5559 “ cima 5560 suc csuc 6195 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ωcom 7582 reccrdg 8047 1c1 10540 + caddc 10542 ℕcn 11640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-nn 11641 |
This theorem is referenced by: dfnn2 11653 dfnn3 11654 peano2nnd 11657 nnind 11658 nnaddcl 11663 2nn 11713 3nn 11719 4nn 11723 5nn 11726 6nn 11729 7nn 11732 8nn 11735 9nn 11738 nnunb 11896 nneo 12069 10nn 12117 fzonn0p1p1 13119 ser1const 13429 expp1 13439 facp1 13641 relexpsucnnl 14393 isercolllem1 15023 isercoll2 15027 climcndslem2 15207 climcnds 15208 harmonic 15216 trireciplem 15219 trirecip 15220 rpnnen2lem9 15577 sqrt2irr 15604 nno 15735 nnoddm1d2 15739 rplpwr 15909 prmind2 16031 eulerthlem2 16121 pcmpt 16230 pockthi 16245 prmreclem6 16259 dec5nprm 16404 mulgnnp1 18238 chfacfisf 21464 chfacfisfcpmat 21465 cayhamlem1 21476 1stcfb 22055 bcthlem3 23931 bcthlem4 23932 ovolunlem1a 24099 ovolicc2lem4 24123 voliunlem1 24153 volsup 24159 volsup2 24208 itg1climres 24317 mbfi1fseqlem5 24322 itg2monolem1 24353 itg2i1fseqle 24357 itg2i1fseq 24358 itg2i1fseq2 24359 itg2addlem 24361 itg2gt0 24363 itg2cnlem1 24364 aaliou3lem7 24940 emcllem1 25575 emcllem2 25576 emcllem3 25577 emcllem5 25579 emcllem6 25580 emcllem7 25581 zetacvg 25594 lgam1 25643 bclbnd 25858 bposlem5 25866 2sqlem10 26006 dchrisumlem2 26068 logdivbnd 26134 pntrsumo1 26143 pntrsumbnd 26144 wwlksext2clwwlk 27838 numclwwlk2lem1 28157 numclwlk2lem2f 28158 opsqrlem5 29923 opsqrlem6 29924 nnindf 30537 psgnfzto1st 30749 esumpmono 31340 fibp1 31661 rrvsum 31714 subfacp1lem6 32434 subfaclim 32437 bcprod 32972 bccolsum 32973 iprodgam 32976 faclimlem1 32977 faclimlem2 32978 faclim2 32982 nn0prpwlem 33672 mblfinlem2 34932 volsupnfl 34939 seqpo 35024 incsequz 35025 incsequz2 35026 geomcau 35036 heiborlem6 35096 bfplem1 35102 jm2.27dlem4 39616 nnsplit 41633 sumnnodd 41918 stoweidlem20 42312 wallispilem4 42360 wallispi2lem1 42363 wallispi2lem2 42364 stirlinglem4 42369 stirlinglem8 42373 stirlinglem11 42376 stirlinglem12 42377 stirlinglem13 42378 vonioolem2 42970 vonicclem2 42973 deccarry 43518 iccpartres 43585 iccelpart 43600 odz2prm2pw 43732 fmtnoprmfac1 43734 fmtnoprmfac2 43736 lighneallem4 43782 |
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