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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 8756 | . 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 1481 (class class class)co 5782 1c1 7645 + caddc 7647 ℕcn 8744 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 df-inn 8745 |
This theorem is referenced by: exp3vallem 10325 bcpasc 10544 caucvgre 10785 resqrexlemdecn 10816 cvgratnnlemmn 11326 cvgratnnlemseq 11327 cvgratnnlemabsle 11328 eftlub 11433 eirraplem 11519 oddennn 11941 exmidunben 11975 cvgcmp2nlemabs 13402 trilpolemeq1 13408 trilpolemlt1 13409 |
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