Theorem 0npi 10842

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 2730	. 2 ⊢ ∅ = ∅
2		elni 10836	. . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
3	2	simprbi 496	. . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅)
4	3	necon2bi 2956	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N)
5	1, 4	ax-mp 5	1 ⊢ ¬ ∅ ∈ N

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∅c0 4299  ωcom 7845  Ncnpi 10804

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-v 3452  df-dif 3920  df-sn 4593  df-ni 10832

This theorem is referenced by:  addasspi  10855  mulasspi  10857  distrpi  10858  addcanpi  10859  mulcanpi  10860  addnidpi  10861  ltapi  10863  ltmpi  10864  ordpipq  10902