Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3234 | . 2 ⊢ ℕ ⊆ (ℕ ∪ {0}) | |
2 | df-n0 8971 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
3 | 1, 2 | sseqtrri 3127 | 1 ⊢ ℕ ⊆ ℕ0 |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3064 ⊆ wss 3066 {csn 3522 0cc0 7613 ℕcn 8713 ℕ0cn0 8970 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-n0 8971 |
This theorem is referenced by: nnnn0 8977 nnnn0d 9023 expcnv 11266 oddge22np1 11567 |
Copyright terms: Public domain | W3C validator |