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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 7944 | . 2 ⊢ 1 ∈ ℝ | |
2 | peano2re 8080 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ) | |
3 | 2 | rgen 2530 | . 2 ⊢ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ |
4 | peano5nni 8908 | . 2 ⊢ ((1 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ) → ℕ ⊆ ℝ) | |
5 | 1, 3, 4 | mp2an 426 | 1 ⊢ ℕ ⊆ ℝ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 ∀wral 2455 ⊆ wss 3129 (class class class)co 5869 ℝcr 7798 1c1 7800 + caddc 7802 ℕcn 8905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4118 ax-cnex 7890 ax-resscn 7891 ax-1re 7893 ax-addrcl 7896 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-in 3135 df-ss 3142 df-int 3843 df-inn 8906 |
This theorem is referenced by: nnsscn 8910 nnre 8912 nnred 8918 nn0ssre 9166 nninfdclemp1 12431 nninfdclemf1 12433 |
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