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Theorem nn0ssre 8242
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 8239 . 2 0 = (ℕ ∪ {0})
2 nnssre 7993 . . 3 ℕ ⊆ ℝ
3 0re 7084 . . . 4 0 ∈ ℝ
4 snssi 3535 . . . 4 (0 ∈ ℝ → {0} ⊆ ℝ)
53, 4ax-mp 7 . . 3 {0} ⊆ ℝ
62, 5unssi 3145 . 2 (ℕ ∪ {0}) ⊆ ℝ
71, 6eqsstri 3002 1 0 ⊆ ℝ
Colors of variables: wff set class
Syntax hints:  wcel 1409  cun 2942  wss 2944  {csn 3402  cr 6945  0cc0 6946  cn 7989  0cn0 8238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-cnex 7032  ax-resscn 7033  ax-1re 7035  ax-addrcl 7038  ax-rnegex 7050
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-int 3643  df-inn 7990  df-n0 8239
This theorem is referenced by:  nn0sscn  8243  nn0re  8247  nn0rei  8249  nn0red  8292
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