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| 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 2737 | . 2 ⊢ ∅ = ∅ | |
| 2 | elni 10916 | . . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅)) | |
| 3 | 2 | simprbi 496 | . . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅) | 
| 4 | 3 | necon2bi 2971 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N) | 
| 5 | 1, 4 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ N | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 ωcom 7887 Ncnpi 10884 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-sn 4627 df-ni 10912 | 
| This theorem is referenced by: addasspi 10935 mulasspi 10937 distrpi 10938 addcanpi 10939 mulcanpi 10940 addnidpi 10941 ltapi 10943 ltmpi 10944 ordpipq 10982 | 
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