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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 2735 | . 2 ⊢ ∅ = ∅ | |
2 | elni 10914 | . . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅)) | |
3 | 2 | simprbi 496 | . . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅) |
4 | 3 | necon2bi 2969 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ N |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 ωcom 7887 Ncnpi 10882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-sn 4632 df-ni 10910 |
This theorem is referenced by: addasspi 10933 mulasspi 10935 distrpi 10936 addcanpi 10937 mulcanpi 10938 addnidpi 10939 ltapi 10941 ltmpi 10942 ordpipq 10980 |
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