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Theorem nnssre 8831
 Description: The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
Assertion
Ref Expression
nnssre ℕ ⊆ ℝ

Proof of Theorem nnssre
StepHypRef Expression
1 1re 7871 . 2 1 ∈ ℝ
2 peano2re 8005 . . 3 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
32rgen 2510 . 2 𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ
4 peano5nni 8830 . 2 ((1 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ) → ℕ ⊆ ℝ)
51, 3, 4mp2an 423 1 ℕ ⊆ ℝ
 Colors of variables: wff set class Syntax hints:   ∈ wcel 2128  ∀wral 2435   ⊆ wss 3102  (class class class)co 5821  ℝcr 7725  1c1 7727   + caddc 7729  ℕcn 8827 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-sep 4082  ax-cnex 7817  ax-resscn 7818  ax-1re 7820  ax-addrcl 7823 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-int 3808  df-inn 8828 This theorem is referenced by:  nnsscn  8832  nnre  8834  nnred  8840  nn0ssre  9088
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