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Mirrors > Home > ILE Home > Th. List > nnssre | GIF version |
Description: The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.) |
Ref | Expression |
---|---|
nnssre | ⊢ ℕ ⊆ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 7765 | . 2 ⊢ 1 ∈ ℝ | |
2 | peano2re 7898 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ) | |
3 | 2 | rgen 2485 | . 2 ⊢ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ |
4 | peano5nni 8723 | . 2 ⊢ ((1 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ) → ℕ ⊆ ℝ) | |
5 | 1, 3, 4 | mp2an 422 | 1 ⊢ ℕ ⊆ ℝ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 ∀wral 2416 ⊆ wss 3071 (class class class)co 5774 ℝcr 7619 1c1 7621 + caddc 7623 ℕcn 8720 |
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 ax-sep 4046 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 |
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-ral 2421 df-v 2688 df-in 3077 df-ss 3084 df-int 3772 df-inn 8721 |
This theorem is referenced by: nnsscn 8725 nnre 8727 nnred 8733 nn0ssre 8981 |
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