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Theorem 0npi 10026
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 2825 . 2 ∅ = ∅
2 elni 10020 . . . 4 (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
32simprbi 492 . . 3 (∅ ∈ N → ∅ ≠ ∅)
43necon2bi 3029 . 2 (∅ = ∅ → ¬ ∅ ∈ N)
51, 4ax-mp 5 1 ¬ ∅ ∈ N
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1656  wcel 2164  wne 2999  c0 4146  ωcom 7331  Ncnpi 9988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-v 3416  df-dif 3801  df-sn 4400  df-ni 10016
This theorem is referenced by:  addasspi  10039  mulasspi  10041  distrpi  10042  addcanpi  10043  mulcanpi  10044  addnidpi  10045  ltapi  10047  ltmpi  10048  ordpipq  10086
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