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Theorem peano2nnd 8330
Description: Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
Hypothesis
Ref Expression
nnred.1 (𝜑𝐴 ∈ ℕ)
Assertion
Ref Expression
peano2nnd (𝜑 → (𝐴 + 1) ∈ ℕ)

Proof of Theorem peano2nnd
StepHypRef Expression
1 nnred.1 . 2 (𝜑𝐴 ∈ ℕ)
2 peano2nn 8327 . 2 (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
31, 2syl 14 1 (𝜑 → (𝐴 + 1) ∈ ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  (class class class)co 5590  1c1 7253   + caddc 7255  cn 8315
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-cnex 7338  ax-resscn 7339  ax-1re 7341  ax-addrcl 7344
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-iota 4933  df-fv 4976  df-ov 5593  df-inn 8316
This theorem is referenced by:  expivallem  9792  bcpasc  10008  caucvgre  10240  resqrexlemdecn  10271  oddennn  10984
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