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Theorem 7nn 8854
Description: 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
7nn 7 ∈ ℕ

Proof of Theorem 7nn
StepHypRef Expression
1 df-7 8752 . 2 7 = (6 + 1)
2 6nn 8853 . . 3 6 ∈ ℕ
3 peano2nn 8700 . . 3 (6 ∈ ℕ → (6 + 1) ∈ ℕ)
42, 3ax-mp 5 . 2 (6 + 1) ∈ ℕ
51, 4eqeltri 2190 1 7 ∈ ℕ
Colors of variables: wff set class
Syntax hints:  wcel 1465  (class class class)co 5742  1c1 7589   + caddc 7591  cn 8688  6c6 8743  7c7 8744
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  df-2 8747  df-3 8748  df-4 8749  df-5 8750  df-6 8751  df-7 8752
This theorem is referenced by:  8nn  8855  7nn0  8967
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