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Theorem 0npi 9701
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 2621 . 2 ∅ = ∅
2 elni 9695 . . . 4 (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
32simprbi 480 . . 3 (∅ ∈ N → ∅ ≠ ∅)
43necon2bi 2823 . 2 (∅ = ∅ → ¬ ∅ ∈ N)
51, 4ax-mp 5 1 ¬ ∅ ∈ N
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1482  wcel 1989  wne 2793  c0 3913  ωcom 7062  Ncnpi 9663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-v 3200  df-dif 3575  df-sn 4176  df-ni 9691
This theorem is referenced by:  addasspi  9714  mulasspi  9716  distrpi  9717  addcanpi  9718  mulcanpi  9719  addnidpi  9720  ltapi  9722  ltmpi  9723  ordpipq  9761
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