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Theorem nn0ssre 9178
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 9175 . 2 0 = (ℕ ∪ {0})
2 nnssre 8921 . . 3 ℕ ⊆ ℝ
3 0re 7956 . . . 4 0 ∈ ℝ
4 snssi 3736 . . . 4 (0 ∈ ℝ → {0} ⊆ ℝ)
53, 4ax-mp 5 . . 3 {0} ⊆ ℝ
62, 5unssi 3310 . 2 (ℕ ∪ {0}) ⊆ ℝ
71, 6eqsstri 3187 1 0 ⊆ ℝ
Colors of variables: wff set class
Syntax hints:  wcel 2148  cun 3127  wss 3129  {csn 3592  cr 7809  0cc0 7810  cn 8917  0cn0 9174
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 4121  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907  ax-rnegex 7919
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-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-int 3845  df-inn 8918  df-n0 9175
This theorem is referenced by:  nn0sscn  9179  nn0re  9183  nn0rei  9185  nn0red  9228
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