![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2726 | . 2 ⊢ ∅ = ∅ | |
2 | elni 10919 | . . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅)) | |
3 | 2 | simprbi 495 | . . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅) |
4 | 3 | necon2bi 2961 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ N |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∅c0 4325 ωcom 7876 Ncnpi 10887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-v 3464 df-dif 3950 df-sn 4634 df-ni 10915 |
This theorem is referenced by: addasspi 10938 mulasspi 10940 distrpi 10941 addcanpi 10942 mulcanpi 10943 addnidpi 10944 ltapi 10946 ltmpi 10947 ordpipq 10985 |
Copyright terms: Public domain | W3C validator |