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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 9573 | . . 3 ⊢ N = (ω ∖ {∅}) | |
2 | difss 3699 | . . 3 ⊢ (ω ∖ {∅}) ⊆ ω | |
3 | 1, 2 | eqsstri 3598 | . 2 ⊢ N ⊆ ω |
4 | 3 | sseli 3564 | 1 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∖ cdif 3537 ∅c0 3874 {csn 4125 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-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-ni 9573 |
This theorem is referenced by: pion 9580 piord 9581 mulidpi 9587 addclpi 9593 mulclpi 9594 addcompi 9595 addasspi 9596 mulcompi 9597 mulasspi 9598 distrpi 9599 addcanpi 9600 mulcanpi 9601 addnidpi 9602 ltexpi 9603 ltapi 9604 ltmpi 9605 indpi 9608 |
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