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Mirrors > Home > ILE Home > Th. List > 0npi | GIF version |
Description: The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) |
Ref | Expression |
---|---|
0npi | ⊢ ¬ ∅ ∈ N |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2081 | . 2 ⊢ ∅ = ∅ | |
2 | elni 6498 | . . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅)) | |
3 | 2 | simprbi 269 | . . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅) |
4 | 3 | necon2bi 2300 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N) |
5 | 1, 4 | ax-mp 7 | 1 ⊢ ¬ ∅ ∈ N |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1284 ∈ wcel 1433 ≠ wne 2245 ∅c0 3251 ωcom 4331 Ncnpi 6462 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-v 2603 df-dif 2975 df-sn 3404 df-ni 6494 |
This theorem is referenced by: elni2 6504 |
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