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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 8859 | . 2 ⊢ 1 ∈ ℕ | |
2 | nn0nnaddcl 9136 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ∈ ℕ) → (𝑁 + 1) ∈ ℕ) | |
3 | 1, 2 | mpan2 422 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 (class class class)co 5836 1c1 7745 + caddc 7747 ℕcn 8848 ℕ0cn0 9105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-iota 5147 df-fv 5190 df-ov 5839 df-inn 8849 df-n0 9106 |
This theorem is referenced by: elnn0nn 9147 elz2 9253 peano5uzti 9290 fseq1p1m1 10019 fzonn0p1 10136 nn0ennn 10358 faccl 10637 facdiv 10640 facwordi 10642 faclbnd 10643 facubnd 10647 bcm1k 10662 bcp1n 10663 bcp1nk 10664 bcpasc 10668 bcxmas 11416 efcllemp 11585 prmfac1 12063 pcfac 12259 |
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