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Mirrors > Home > MPE Home > Th. List > 0npi | Structured version Visualization version GIF version |
Description: The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0npi | ⊢ ¬ ∅ ∈ N |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . 2 ⊢ ∅ = ∅ | |
2 | elni 10286 | . . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅)) | |
3 | 2 | simprbi 497 | . . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅) |
4 | 3 | necon2bi 3043 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ N |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 ωcom 7569 Ncnpi 10254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-v 3494 df-dif 3936 df-sn 4558 df-ni 10282 |
This theorem is referenced by: addasspi 10305 mulasspi 10307 distrpi 10308 addcanpi 10309 mulcanpi 10310 addnidpi 10311 ltapi 10313 ltmpi 10314 ordpipq 10352 |
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