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Theorem peano2nnd 8936
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  |-  ( ph  ->  A  e.  NN )
Assertion
Ref Expression
peano2nnd  |-  ( ph  ->  ( A  +  1 )  e.  NN )

Proof of Theorem peano2nnd
StepHypRef Expression
1 nnred.1 . 2  |-  ( ph  ->  A  e.  NN )
2 peano2nn 8933 . 2  |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
31, 2syl 14 1  |-  ( ph  ->  ( A  +  1 )  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2148  (class class class)co 5877   1c1 7814    + caddc 7816   NNcn 8921
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4123  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880  df-inn 8922
This theorem is referenced by:  exp3vallem  10523  bcpasc  10748  caucvgre  10992  resqrexlemdecn  11023  cvgratnnlemmn  11535  cvgratnnlemseq  11536  cvgratnnlemabsle  11537  eftlub  11700  eirraplem  11786  infpnlem1  12359  infpnlem2  12360  1arith  12367  oddennn  12395  exmidunben  12429  nninfdclemp1  12453  nninfdclemlt  12454  lgsdilem2  14522  cvgcmp2nlemabs  14865  trilpolemeq1  14873  trilpolemlt1  14874  nconstwlpolemgt0  14897
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