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Theorem 0npi 10922
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 2737 . 2 ∅ = ∅
2 elni 10916 . . . 4 (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
32simprbi 496 . . 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 2108  wne 2940  c0 4333  ωcom 7887  Ncnpi 10884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-sn 4627  df-ni 10912
This theorem is referenced by:  addasspi  10935  mulasspi  10937  distrpi  10938  addcanpi  10939  mulcanpi  10940  addnidpi  10941  ltapi  10943  ltmpi  10944  ordpipq  10982
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