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Theorem nn0ssre 8988
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 8985 . 2 0 = (ℕ ∪ {0})
2 nnssre 8731 . . 3 ℕ ⊆ ℝ
3 0re 7773 . . . 4 0 ∈ ℝ
4 snssi 3664 . . . 4 (0 ∈ ℝ → {0} ⊆ ℝ)
53, 4ax-mp 5 . . 3 {0} ⊆ ℝ
62, 5unssi 3251 . 2 (ℕ ∪ {0}) ⊆ ℝ
71, 6eqsstri 3129 1 0 ⊆ ℝ
Colors of variables: wff set class
Syntax hints:  wcel 1480  cun 3069  wss 3071  {csn 3527  cr 7626  0cc0 7627  cn 8727  0cn0 8984
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-cnex 7718  ax-resscn 7719  ax-1re 7721  ax-addrcl 7724  ax-rnegex 7736
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-int 3772  df-inn 8728  df-n0 8985
This theorem is referenced by:  nn0sscn  8989  nn0re  8993  nn0rei  8995  nn0red  9038
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