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Mirrors > Home > MPE Home > Th. List > pinn | Structured version Visualization version GIF version |
Description: A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pinn | ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ni 10910 | . . 3 ⊢ N = (ω ∖ {∅}) | |
2 | difss 4146 | . . 3 ⊢ (ω ∖ {∅}) ⊆ ω | |
3 | 1, 2 | eqsstri 4030 | . 2 ⊢ N ⊆ ω |
4 | 3 | sseli 3991 | 1 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∖ cdif 3960 ∅c0 4339 {csn 4631 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-v 3480 df-dif 3966 df-ss 3980 df-ni 10910 |
This theorem is referenced by: pion 10917 piord 10918 mulidpi 10924 addclpi 10930 mulclpi 10931 addcompi 10932 addasspi 10933 mulcompi 10934 mulasspi 10935 distrpi 10936 addcanpi 10937 mulcanpi 10938 addnidpi 10939 ltexpi 10940 ltapi 10941 ltmpi 10942 indpi 10945 |
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