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Theorem nn0ssre 8939
Description: Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
Assertion
Ref Expression
nn0ssre 0 ⊆ ℝ

Proof of Theorem nn0ssre
StepHypRef Expression
1 df-n0 8936 . 2 0 = (ℕ ∪ {0})
2 nnssre 8688 . . 3 ℕ ⊆ ℝ
3 0re 7734 . . . 4 0 ∈ ℝ
4 snssi 3634 . . . 4 (0 ∈ ℝ → {0} ⊆ ℝ)
53, 4ax-mp 5 . . 3 {0} ⊆ ℝ
62, 5unssi 3221 . 2 (ℕ ∪ {0}) ⊆ ℝ
71, 6eqsstri 3099 1 0 ⊆ ℝ
Colors of variables: wff set class
Syntax hints:  wcel 1465  cun 3039  wss 3041  {csn 3497  cr 7587  0cc0 7588  cn 8684  0cn0 8935
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  ax-rnegex 7697
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-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-int 3742  df-inn 8685  df-n0 8936
This theorem is referenced by:  nn0sscn  8940  nn0re  8944  nn0rei  8946  nn0red  8989
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