Theorem 0npi 10894

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 2762	. 2 ⊢ ∅ = ∅
2		elni 10888	. . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
3	2	simprbi 503	. . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅)
4	3	necon2bi 2987	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N)
5	1, 4	ax-mp 5	1 ⊢ ¬ ∅ ∈ N

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∅c0 4282  ωcom 7865  Ncnpi 10856

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3455  df-dif 3905  df-sn 4588  df-ni 10884

This theorem is used by:  addasspi  10907  mulasspi  10909  distrpi  10910  addcanpi  10911  mulcanpi  10912  addnidpi  10913  ltapi  10915  ltmpi  10916  ordpipq  10954