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Theorem nnssre 8692
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 7733 . 2 1 ∈ ℝ
2 peano2re 7866 . . 3 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
32rgen 2462 . 2 𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ
4 peano5nni 8691 . 2 ((1 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ) → ℕ ⊆ ℝ)
51, 3, 4mp2an 422 1 ℕ ⊆ ℝ
Colors of variables: wff set class
Syntax hints:  wcel 1465  wral 2393  wss 3041  (class class class)co 5742  cr 7587  1c1 7589   + caddc 7591  cn 8688
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-in 3047  df-ss 3054  df-int 3742  df-inn 8689
This theorem is referenced by:  nnsscn  8693  nnre  8695  nnred  8701  nn0ssre  8949
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