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

Step	Hyp	Ref	Expression
1		eqid 2825	. 2 ⊢ ∅ = ∅
2		elni 10020	. . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
3	2	simprbi 492	. . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅)
4	3	necon2bi 3029	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N)
5	1, 4	ax-mp 5	1 ⊢ ¬ ∅ ∈ N

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