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Theorem nn0ssre 8647
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 8644 . 2 0 = (ℕ ∪ {0})
2 nnssre 8398 . . 3 ℕ ⊆ ℝ
3 0re 7467 . . . 4 0 ∈ ℝ
4 snssi 3576 . . . 4 (0 ∈ ℝ → {0} ⊆ ℝ)
53, 4ax-mp 7 . . 3 {0} ⊆ ℝ
62, 5unssi 3173 . 2 (ℕ ∪ {0}) ⊆ ℝ
71, 6eqsstri 3054 1 0 ⊆ ℝ
Colors of variables: wff set class
Syntax hints:  wcel 1438  cun 2995  wss 2997  {csn 3441  cr 7328  0cc0 7329  cn 8394  0cn0 8643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-cnex 7415  ax-resscn 7416  ax-1re 7418  ax-addrcl 7421  ax-rnegex 7433
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-int 3684  df-inn 8395  df-n0 8644
This theorem is referenced by:  nn0sscn  8648  nn0re  8652  nn0rei  8654  nn0red  8697
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