Theorem 0npi 10991

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 2763	. 2 ⊢ ∅ = ∅
2		elni 10985	. . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
3	2	simprbi 503	. . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅)
4	3	necon2bi 2989	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N)
5	1, 4	ax-mp 5	1 ⊢ ¬ ∅ ∈ N

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∅c0 4281  ωcom 7898  Ncnpi 10953

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-dif 4050  df-sn 4589  df-ni 10981

This theorem is used by:  addasspi  11004  mulasspi  11006  distrpi  11007  addcanpi  11008  mulcanpi  11009  addnidpi  11010  ltapi  11012  ltmpi  11013  ordpipq  11051