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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 10912 | . . 3 ⊢ N = (ω ∖ {∅}) | |
| 2 | difss 4136 | . . 3 ⊢ (ω ∖ {∅}) ⊆ ω | |
| 3 | 1, 2 | eqsstri 4030 | . 2 ⊢ N ⊆ ω | 
| 4 | 3 | sseli 3979 | 1 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ∖ cdif 3948 ∅c0 4333 {csn 4626 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-v 3482 df-dif 3954 df-ss 3968 df-ni 10912 | 
| This theorem is referenced by: pion 10919 piord 10920 mulidpi 10926 addclpi 10932 mulclpi 10933 addcompi 10934 addasspi 10935 mulcompi 10936 mulasspi 10937 distrpi 10938 addcanpi 10939 mulcanpi 10940 addnidpi 10941 ltexpi 10942 ltapi 10943 ltmpi 10944 indpi 10947 | 
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