ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  peano2nnd Unicode version

Theorem peano2nnd 8703
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 8700 . 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 1465  (class class class)co 5742   1c1 7589    + caddc 7591   NNcn 8688
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745  df-inn 8689
This theorem is referenced by:  exp3vallem  10262  bcpasc  10480  caucvgre  10721  resqrexlemdecn  10752  cvgratnnlemmn  11262  cvgratnnlemseq  11263  cvgratnnlemabsle  11264  eftlub  11323  eirraplem  11410  oddennn  11832  exmidunben  11866  cvgcmp2nlemabs  13154  trilpolemeq1  13160  trilpolemlt1  13161
  Copyright terms: Public domain W3C validator