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Mirrors > Home > ILE Home > Th. List > peano2nnd | GIF version |
Description: Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnred.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
peano2nnd | ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | peano2nn 8732 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 (class class class)co 5774 1c1 7621 + caddc 7623 ℕcn 8720 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-inn 8721 |
This theorem is referenced by: exp3vallem 10294 bcpasc 10512 caucvgre 10753 resqrexlemdecn 10784 cvgratnnlemmn 11294 cvgratnnlemseq 11295 cvgratnnlemabsle 11296 eftlub 11396 eirraplem 11483 oddennn 11905 exmidunben 11939 cvgcmp2nlemabs 13227 trilpolemeq1 13233 trilpolemlt1 13234 |
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