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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 8050 | . 2 ⊢ 1 ∈ ℕ | |
2 | nn0nnaddcl 8319 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ∈ ℕ) → (𝑁 + 1) ∈ ℕ) | |
3 | 1, 2 | mpan2 415 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 (class class class)co 5532 1c1 6982 + caddc 6984 ℕcn 8039 ℕ0cn0 8288 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 df-inn 8040 df-n0 8289 |
This theorem is referenced by: elnn0nn 8330 elz2 8419 peano5uzti 8455 fseq1p1m1 9111 fzonn0p1 9220 nn0ennn 9425 faccl 9662 facdiv 9665 facwordi 9667 faclbnd 9668 facubnd 9672 bcm1k 9687 bcp1n 9688 bcp1nk 9689 bcpasc 9693 prmfac1 10531 |
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