Theorem 0npi 10922

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 2737	. 2 ⊢ ∅ = ∅
2		elni 10916	. . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
3	2	simprbi 496	. . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅)
4	3	necon2bi 2971	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N)
5	1, 4	ax-mp 5	1 ⊢ ¬ ∅ ∈ N

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∅c0 4333  ωcom 7887  Ncnpi 10884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-sn 4627  df-ni 10912

This theorem is referenced by:  addasspi  10935  mulasspi  10937  distrpi  10938  addcanpi  10939  mulcanpi  10940  addnidpi  10941  ltapi  10943  ltmpi  10944  ordpipq  10982