Theorem 0npi 10967

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 2760	. 2 ⊢ ∅ = ∅
2		elni 10961	. . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
3	2	simprbi 503	. . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅)
4	3	necon2bi 2985	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N)
5	1, 4	ax-mp 5	1 ⊢ ¬ ∅ ∈ N

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∅c0 4278  ωcom 7877  Ncnpi 10929

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 4047  df-sn 4584  df-ni 10957

This theorem is used by:  addasspi  10980  mulasspi  10982  distrpi  10983  addcanpi  10984  mulcanpi  10985  addnidpi  10986  ltapi  10988  ltmpi  10989  ordpipq  11027