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Theorem peano2nnd 8535
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 8532 . 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 1445  (class class class)co 5690  1c1 7448   + caddc 7450  cn 8520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-cnex 7533  ax-resscn 7534  ax-1re 7536  ax-addrcl 7539
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-iota 5014  df-fv 5057  df-ov 5693  df-inn 8521
This theorem is referenced by:  exp3vallem  10087  bcpasc  10305  caucvgre  10545  resqrexlemdecn  10576  cvgratnnlemmn  11083  cvgratnnlemseq  11084  cvgratnnlemabsle  11085  eftlub  11144  eirraplem  11228  oddennn  11647
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