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Theorem 0npi 9905
 Description: The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
0npi ¬ ∅ ∈ N

Proof of Theorem 0npi
StepHypRef Expression
1 eqid 2770 . 2 ∅ = ∅
2 elni 9899 . . . 4 (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
32simprbi 478 . . 3 (∅ ∈ N → ∅ ≠ ∅)
43necon2bi 2972 . 2 (∅ = ∅ → ¬ ∅ ∈ N)
51, 4ax-mp 5 1 ¬ ∅ ∈ N
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1630   ∈ wcel 2144   ≠ wne 2942  ∅c0 4061  ωcom 7211  Ncnpi 9867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-v 3351  df-dif 3724  df-sn 4315  df-ni 9895 This theorem is referenced by:  addasspi  9918  mulasspi  9920  distrpi  9921  addcanpi  9922  mulcanpi  9923  addnidpi  9924  ltapi  9926  ltmpi  9927  ordpipq  9965
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