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Mirrors > Home > ILE Home > Th. List > nnssnn0 | GIF version |
Description: Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Ref | Expression |
---|---|
nnssnn0 | ⊢ ℕ ⊆ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 3281 | . 2 ⊢ ℕ ⊆ (ℕ ∪ {0}) | |
2 | df-n0 9107 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
3 | 1, 2 | sseqtrri 3173 | 1 ⊢ ℕ ⊆ ℕ0 |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3110 ⊆ wss 3112 {csn 3571 0cc0 7745 ℕcn 8849 ℕ0cn0 9106 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-n0 9107 |
This theorem is referenced by: nnnn0 9113 nnnn0d 9159 expcnv 11435 oddge22np1 11807 |
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