Theorem 0npi 10293

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 2798	. 2 ⊢ ∅ = ∅
2		elni 10287	. . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
3	2	simprbi 500	. . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅)
4	3	necon2bi 3017	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N)
5	1, 4	ax-mp 5	1 ⊢ ¬ ∅ ∈ N

Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∅c0 4243  ωcom 7560  Ncnpi 10255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-v 3443  df-dif 3884  df-sn 4526  df-ni 10283

This theorem is referenced by:  addasspi  10306  mulasspi  10308  distrpi  10309  addcanpi  10310  mulcanpi  10311  addnidpi  10312  ltapi  10314  ltmpi  10315  ordpipq  10353