Theorem 0npi 6411

Description: The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)

Assertion

Ref	Expression
0npi	⊢ ¬ ∅ ∈ N

Proof of Theorem 0npi

Step	Hyp	Ref	Expression
1		eqid 2040	. 2 ⊢ ∅ = ∅
2		elni 6406	. . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
3	2	simprbi 260	. . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅)
4	3	necon2bi 2260	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N)
5	1, 4	ax-mp 7	1 ⊢ ¬ ∅ ∈ N

Syntax hints:  ¬ wn 3   = wceq 1243   ∈ wcel 1393   ≠ wne 2204  ∅c0 3224  ωcom 4313  Ncnpi 6370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-sn 3381  df-ni 6402

This theorem is referenced by:  elni2  6412