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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
StepHypRef Expression
1 eqid 2040 . 2 ∅ = ∅
2 elni 6406 . . . 4 (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
32simprbi 260 . . 3 (∅ ∈ N → ∅ ≠ ∅)
43necon2bi 2260 . 2 (∅ = ∅ → ¬ ∅ ∈ N)
51, 4ax-mp 7 1 ¬ ∅ ∈ N
 Colors of variables: wff set class 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
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