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Theorem 7nn 8119
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 8024 . 2 7 = (6 + 1)
2 6nn 8118 . . 3 6 ∈ ℕ
3 peano2nn 7972 . . 3 (6 ∈ ℕ → (6 + 1) ∈ ℕ)
42, 3ax-mp 7 . 2 (6 + 1) ∈ ℕ
51, 4eqeltri 2124 1 7 ∈ ℕ
Colors of variables: wff set class
Syntax hints:  wcel 1407  (class class class)co 5537  1c1 6918   + caddc 6920  cn 7960  6c6 8014  7c7 8015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-cnex 7003  ax-resscn 7004  ax-1re 7006  ax-addrcl 7009
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-br 3790  df-iota 4892  df-fv 4935  df-ov 5540  df-inn 7961  df-2 8019  df-3 8020  df-4 8021  df-5 8022  df-6 8023  df-7 8024
This theorem is referenced by:  8nn  8120  7nn0  8231
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