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Theorem peano2nnd 8964
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 8961 . 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 2160  (class class class)co 5896  1c1 7842   + caddc 7844  cn 8949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136  ax-cnex 7932  ax-resscn 7933  ax-1re 7935  ax-addrcl 7938
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5899  df-inn 8950
This theorem is referenced by:  exp3vallem  10552  bcpasc  10778  caucvgre  11022  resqrexlemdecn  11053  cvgratnnlemmn  11565  cvgratnnlemseq  11566  cvgratnnlemabsle  11567  eftlub  11730  eirraplem  11816  infpnlem1  12391  infpnlem2  12392  1arith  12399  oddennn  12443  exmidunben  12477  nninfdclemp1  12501  nninfdclemlt  12502  lgsdilem2  14895  cvgcmp2nlemabs  15239  trilpolemeq1  15247  trilpolemlt1  15248  nconstwlpolemgt0  15271
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