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Mirrors > Home > ILE Home > Th. List > nnsscn | GIF version |
Description: The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
Ref | Expression |
---|---|
nnsscn | ⊢ ℕ ⊆ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssre 8748 | . 2 ⊢ ℕ ⊆ ℝ | |
2 | ax-resscn 7736 | . 2 ⊢ ℝ ⊆ ℂ | |
3 | 1, 2 | sstri 3111 | 1 ⊢ ℕ ⊆ ℂ |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3076 ℂcc 7642 ℝcr 7643 ℕcn 8744 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-in 3082 df-ss 3089 df-int 3780 df-inn 8745 |
This theorem is referenced by: nnex 8750 nncn 8752 nncnd 8758 nn0addcl 9036 nn0mulcl 9037 dfz2 9147 nnexpcl 10337 |
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