Theorem elni 10828

Description: Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
elni	⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Proof of Theorem elni

Step	Hyp	Ref	Expression
1		df-ni 10824	. . 3 ⊢ N = (ω ∖ {∅})
2	1	eleq2i 2853	. 2 ⊢ (𝐴 ∈ N ↔ 𝐴 ∈ (ω ∖ {∅}))
3		eldifsn 4743	. 2 ⊢ (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
4	2, 3	bitri 277	1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2141   ≠ wne 2956   ∖ cdif 3899  ∅c0 4283  {csn 4579  ωcom 7841  Ncnpi 10796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-dif 3905  df-sn 4580  df-ni 10824

This theorem is referenced by:  elni2  10829  0npi  10834  1pi  10835  addclpi  10844  mulclpi  10845  nlt1pi  10858  indpi  10859