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Theorem 0npi 10739
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 2736 . 2 ∅ = ∅
2 elni 10733 . . . 4 (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
32simprbi 497 . . 3 (∅ ∈ N → ∅ ≠ ∅)
43necon2bi 2971 . 2 (∅ = ∅ → ¬ ∅ ∈ N)
51, 4ax-mp 5 1 ¬ ∅ ∈ N
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2105  wne 2940  c0 4269  ωcom 7780  Ncnpi 10701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-v 3443  df-dif 3901  df-sn 4574  df-ni 10729
This theorem is referenced by:  addasspi  10752  mulasspi  10754  distrpi  10755  addcanpi  10756  mulcanpi  10757  addnidpi  10758  ltapi  10760  ltmpi  10761  ordpipq  10799
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