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Theorem nnssre 7918
 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 7026 . 2 1 ∈ ℝ
2 peano2re 7149 . . 3 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
32rgen 2374 . 2 𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ
4 peano5nni 7917 . 2 ((1 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ) → ℕ ⊆ ℝ)
51, 3, 4mp2an 402 1 ℕ ⊆ ℝ
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393  ∀wral 2306   ⊆ wss 2917  (class class class)co 5512  ℝcr 6888  1c1 6890   + caddc 6892  ℕcn 7914 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-cnex 6975  ax-resscn 6976  ax-1re 6978  ax-addrcl 6981 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-int 3616  df-inn 7915 This theorem is referenced by:  nnsscn  7919  nnre  7921  nnred  7927  nn0ssre  8185
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