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Theorem 0npi 10292
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 2818 . 2 ∅ = ∅
2 elni 10286 . . . 4 (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
32simprbi 497 . . 3 (∅ ∈ N → ∅ ≠ ∅)
43necon2bi 3043 . 2 (∅ = ∅ → ¬ ∅ ∈ N)
51, 4ax-mp 5 1 ¬ ∅ ∈ N
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1528  wcel 2105  wne 3013  c0 4288  ωcom 7569  Ncnpi 10254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-v 3494  df-dif 3936  df-sn 4558  df-ni 10282
This theorem is referenced by:  addasspi  10305  mulasspi  10307  distrpi  10308  addcanpi  10309  mulcanpi  10310  addnidpi  10311  ltapi  10313  ltmpi  10314  ordpipq  10352
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