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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 7417 | . . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) Fn ω | |
2 | fvelrnb 6153 | . . . 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 6577 | . . . . . . 7 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1) ∈ V | |
5 | eqid 2610 | . . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) | |
6 | oveq1 6556 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 + 1) = (𝑥 + 1)) | |
7 | oveq1 6556 | . . . . . . . 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 7422 | . . . . . . 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 703 | . . . . . 6 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1)) |
10 | peano2 6978 | . . . . . . . 8 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω) | |
11 | fnfvelrn 6264 | . . . . . . . 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 694 | . . . . . . 7 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘suc 𝑦) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)) |
13 | df-nn 10898 | . . . . . . . 8 ⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) | |
14 | df-ima 5051 | . . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) | |
15 | 13, 14 | eqtri 2632 | . . . . . . 7 ⊢ ℕ = ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) |
16 | 12, 15 | syl6eleqr 2699 | . . . . . 6 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘suc 𝑦) ∈ ℕ) |
17 | 9, 16 | eqeltrrd 2689 | . . . . 5 ⊢ (𝑦 ∈ ω → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1) ∈ ℕ) |
18 | oveq1 6556 | . . . . . 6 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) = 𝐴 → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) + 1) = (𝐴 + 1)) | |
19 | 18 | eleq1d 2672 | . . . . 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 234 | . . . 4 ⊢ (𝑦 ∈ ω → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) = 𝐴 → (𝐴 + 1) ∈ ℕ)) |
21 | 20 | rexlimiv 3009 | . . 3 ⊢ (∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)‘𝑦) = 𝐴 → (𝐴 + 1) ∈ ℕ) |
22 | 3, 21 | sylbi 206 | . 2 ⊢ (𝐴 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω) → (𝐴 + 1) ∈ ℕ) |
23 | 22, 15 | eleq2s 2706 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ↦ cmpt 4643 ran crn 5039 ↾ cres 5040 “ cima 5041 suc csuc 5642 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 ωcom 6957 reccrdg 7392 1c1 9816 + caddc 9818 ℕcn 10897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 |
This theorem is referenced by: dfnn2 10910 dfnn3 10911 peano2nnd 10914 nnind 10915 nnaddcl 10919 2nn 11062 3nn 11063 4nn 11064 5nn 11065 6nn 11066 7nn 11067 8nn 11068 9nn 11069 10nnOLD 11070 nnunb 11165 nneo 11337 10nn 11390 fzonn0p1p1 12413 ser1const 12719 expp1 12729 facp1 12927 relexpsucnnl 13620 isercolllem1 14243 isercoll2 14247 climcndslem2 14421 climcnds 14422 harmonic 14430 trireciplem 14433 trirecip 14434 rpnnen2lem9 14790 sqrt2irr 14817 nno 14936 nnoddm1d2 14940 rplpwr 15114 prmind2 15236 eulerthlem2 15325 pcmpt 15434 pockthi 15449 prmreclem6 15463 dec5nprm 15608 mulgnnp1 17372 chfacfisf 20478 chfacfisfcpmat 20479 cayhamlem1 20490 1stcfb 21058 bcthlem3 22931 bcthlem4 22932 ovolunlem1a 23071 ovolicc2lem4 23095 voliunlem1 23125 volsup 23131 volsup2 23179 itg1climres 23287 mbfi1fseqlem5 23292 itg2monolem1 23323 itg2i1fseqle 23327 itg2i1fseq 23328 itg2i1fseq2 23329 itg2addlem 23331 itg2gt0 23333 itg2cnlem1 23334 aaliou3lem7 23908 emcllem1 24522 emcllem2 24523 emcllem3 24524 emcllem5 24526 emcllem6 24527 emcllem7 24528 zetacvg 24541 lgam1 24590 bclbnd 24805 bposlem5 24813 2sqlem10 24953 dchrisumlem2 24979 logdivbnd 25045 pntrsumo1 25054 pntrsumbnd 25055 wwlkext2clwwlk 26331 numclwwlk2lem1 26629 numclwlk2lem2f 26630 opsqrlem5 28387 opsqrlem6 28388 nnindf 28952 psgnfzto1st 29186 esumpmono 29468 fibp1 29790 rrvsum 29843 subfacp1lem6 30421 subfaclim 30424 bcprod 30877 bccolsum 30878 iprodgam 30881 faclimlem1 30882 faclimlem2 30883 faclim2 30887 nn0prpwlem 31487 mblfinlem2 32617 volsupnfl 32624 seqpo 32713 incsequz 32714 incsequz2 32715 geomcau 32725 heiborlem6 32785 bfplem1 32791 jm2.27dlem4 36597 nnsplit 38515 sumnnodd 38697 stoweidlem20 38913 wallispilem4 38961 wallispi2lem1 38964 wallispi2lem2 38965 stirlinglem4 38970 stirlinglem8 38974 stirlinglem11 38977 stirlinglem12 38978 stirlinglem13 38979 vonioolem2 39572 vonicclem2 39575 deccarry 39941 iccpartres 39956 iccelpart 39971 odz2prm2pw 40013 fmtnoprmfac1 40015 fmtnoprmfac2 40017 lighneallem4 40065 wwlksext2clwwlk 41231 av-numclwwlk2lem1 41532 av-numclwlk2lem2f 41533 |
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