Theorem 0npi 10306

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 2823	. 2 ⊢ ∅ = ∅
2		elni 10300	. . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
3	2	simprbi 499	. . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅)
4	3	necon2bi 3048	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N)
5	1, 4	ax-mp 5	1 ⊢ ¬ ∅ ∈ N

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∅c0 4293  ωcom 7582  Ncnpi 10268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-v 3498  df-dif 3941  df-sn 4570  df-ni 10296

This theorem is referenced by:  addasspi  10319  mulasspi  10321  distrpi  10322  addcanpi  10323  mulcanpi  10324  addnidpi  10325  ltapi  10327  ltmpi  10328  ordpipq  10366