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Mirrors > Home > ILE Home > Th. List > peano2nn0 | GIF version |
Description: Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
peano2nn0 | ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 9206 | . 2 ⊢ 1 ∈ ℕ0 | |
2 | nn0addcl 9225 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ0) | |
3 | 1, 2 | mpan2 425 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2158 (class class class)co 5888 1c1 7826 + caddc 7828 ℕ0cn0 9190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-sep 4133 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-i2m1 7930 ax-0id 7933 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-iota 5190 df-fv 5236 df-ov 5891 df-inn 8934 df-n0 9191 |
This theorem is referenced by: peano2z 9303 nn0split 10150 fzonn0p1p1 10227 elfzom1p1elfzo 10228 frecfzennn 10440 leexp2r 10588 facdiv 10732 facwordi 10734 faclbnd 10735 faclbnd2 10736 faclbnd3 10737 faclbnd6 10738 bcnp1n 10753 bcp1m1 10759 bcpasc 10760 hashfz 10815 bcxmas 11511 geolim 11533 geo2sum 11536 mertenslemub 11556 mertenslemi1 11557 mertenslem2 11558 mertensabs 11559 efcllemp 11680 eftlub 11712 efsep 11713 effsumlt 11714 nn0ob 11927 nn0oddm1d2 11928 nn0seqcvgd 12055 algcvg 12062 pw2dvdseulemle 12181 2sqpwodd 12190 nonsq 12221 pcprendvds 12304 pcpremul 12307 pcdvdsb 12333 ennnfonelemp1 12421 ennnfonelemkh 12427 ennnfonelemim 12439 |
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