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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 2610 | . 2 ⊢ ∅ = ∅ | |
2 | elni 9577 | . . . 4 ⊢ (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅)) | |
3 | 2 | simprbi 479 | . . 3 ⊢ (∅ ∈ N → ∅ ≠ ∅) |
4 | 3 | necon2bi 2812 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ N) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ N |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 ωcom 6957 Ncnpi 9545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-sn 4126 df-ni 9573 |
This theorem is referenced by: addasspi 9596 mulasspi 9598 distrpi 9599 addcanpi 9600 mulcanpi 9601 addnidpi 9602 ltapi 9604 ltmpi 9605 ordpipq 9643 |
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