Theorem 0npi 10801

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 2741	. 2 ⊢ ∅ = ∅
2		elni 10795	. . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
3	2	simprbi 499	. . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅)
4	3	necon2bi 2966	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N)
5	1, 4	ax-mp 5	1 ⊢ ¬ ∅ ∈ N

Syntax hints:  ¬ wn 3   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∅c0 4263  ωcom 7809  Ncnpi 10763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-v 3435  df-dif 3887  df-sn 4558  df-ni 10791

This theorem is referenced by:  addasspi  10814  mulasspi  10816  distrpi  10817  addcanpi  10818  mulcanpi  10819  addnidpi  10820  ltapi  10822  ltmpi  10823  ordpipq  10861