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Mirrors > Home > ILE Home > Th. List > nnnn0i | GIF version |
Description: A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.) |
Ref | Expression |
---|---|
nnnn0.1 | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
nnnn0i | ⊢ 𝑁 ∈ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0.1 | . 2 ⊢ 𝑁 ∈ ℕ | |
2 | nnnn0 8984 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝑁 ∈ ℕ0 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 ℕcn 8720 ℕ0cn0 8977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-n0 8978 |
This theorem is referenced by: 1nn0 8993 2nn0 8994 3nn0 8995 4nn0 8996 5nn0 8997 6nn0 8998 7nn0 8999 8nn0 9000 9nn0 9001 numlt 9206 declei 9217 numlti 9218 |
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