Theorem 0npi 10873

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 10867	. . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
3	2	simprbi 502	. . 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 1569   ∈ wcel 2142   ≠ wne 2957  ∅c0 4285  ωcom 7860  Ncnpi 10835

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456  df-dif 3907  df-sn 4589  df-ni 10863

This theorem is used by:  addasspi  10886  mulasspi  10888  distrpi  10889  addcanpi  10890  mulcanpi  10891  addnidpi  10892  ltapi  10894  ltmpi  10895  ordpipq  10933