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Theorem nnssre 8909
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 7944 . 2 1 ∈ ℝ
2 peano2re 8080 . . 3 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
32rgen 2530 . 2 𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ
4 peano5nni 8908 . 2 ((1 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ) → ℕ ⊆ ℝ)
51, 3, 4mp2an 426 1 ℕ ⊆ ℝ
Colors of variables: wff set class
Syntax hints:  wcel 2148  wral 2455  wss 3129  (class class class)co 5869  cr 7798  1c1 7800   + caddc 7802  cn 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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