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Theorem 0npi 10920
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 2735 . 2 ∅ = ∅
2 elni 10914 . . . 4 (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
32simprbi 496 . . 3 (∅ ∈ N → ∅ ≠ ∅)
43necon2bi 2969 . 2 (∅ = ∅ → ¬ ∅ ∈ N)
51, 4ax-mp 5 1 ¬ ∅ ∈ N
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  wne 2938  c0 4339  ωcom 7887  Ncnpi 10882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-sn 4632  df-ni 10910
This theorem is referenced by:  addasspi  10933  mulasspi  10935  distrpi  10936  addcanpi  10937  mulcanpi  10938  addnidpi  10939  ltapi  10941  ltmpi  10942  ordpipq  10980
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