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Theorem 0npi 10925
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 2726 . 2 ∅ = ∅
2 elni 10919 . . . 4 (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
32simprbi 495 . . 3 (∅ ∈ N → ∅ ≠ ∅)
43necon2bi 2961 . 2 (∅ = ∅ → ¬ ∅ ∈ N)
51, 4ax-mp 5 1 ¬ ∅ ∈ N
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  wcel 2099  wne 2930  c0 4325  ωcom 7876  Ncnpi 10887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-v 3464  df-dif 3950  df-sn 4634  df-ni 10915
This theorem is referenced by:  addasspi  10938  mulasspi  10940  distrpi  10941  addcanpi  10942  mulcanpi  10943  addnidpi  10944  ltapi  10946  ltmpi  10947  ordpipq  10985
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