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Theorem 8nn 9001
Description: 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
8nn 8 ∈ ℕ

Proof of Theorem 8nn
StepHypRef Expression
1 df-8 8899 . 2 8 = (7 + 1)
2 7nn 9000 . . 3 7 ∈ ℕ
3 peano2nn 8846 . . 3 (7 ∈ ℕ → (7 + 1) ∈ ℕ)
42, 3ax-mp 5 . 2 (7 + 1) ∈ ℕ
51, 4eqeltri 2230 1 8 ∈ ℕ
Colors of variables: wff set class
Syntax hints:  wcel 2128  (class class class)co 5825  1c1 7734   + caddc 7736  cn 8834  7c7 8890  8c8 8891
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-sep 4083  ax-cnex 7824  ax-resscn 7825  ax-1re 7827  ax-addrcl 7830
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-br 3967  df-iota 5136  df-fv 5179  df-ov 5828  df-inn 8835  df-2 8893  df-3 8894  df-4 8895  df-5 8896  df-6 8897  df-7 8898  df-8 8899
This theorem is referenced by:  9nn  9002  8nn0  9114  ipndx  12370  ipid  12371  ipslid  12372  ipsstrd  12373
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