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Mirrors > Home > ILE Home > Th. List > nn0p1nn | GIF version |
Description: A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nn0p1nn | ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 8926 | . 2 ⊢ 1 ∈ ℕ | |
2 | nn0nnaddcl 9203 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ∈ ℕ) → (𝑁 + 1) ∈ ℕ) | |
3 | 1, 2 | mpan2 425 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 (class class class)co 5872 1c1 7809 + caddc 7811 ℕcn 8915 ℕ0cn0 9172 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4120 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-i2m1 7913 ax-0id 7916 ax-rnegex 7917 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-iota 5177 df-fv 5223 df-ov 5875 df-inn 8916 df-n0 9173 |
This theorem is referenced by: elnn0nn 9214 elz2 9320 peano5uzti 9357 fseq1p1m1 10089 fzonn0p1 10206 nn0ennn 10428 faccl 10708 facdiv 10711 facwordi 10713 faclbnd 10714 facubnd 10718 bcm1k 10733 bcp1n 10734 bcp1nk 10735 bcpasc 10739 bcxmas 11490 efcllemp 11659 uzwodc 12030 prmfac1 12144 pcfac 12340 |
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