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Theorem nn0ssre 7961
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 7958 . 2 0 = (ℕ ∪ {0})
2 nnssre 7699 . . 3 ℕ ⊆ ℝ
3 0re 6825 . . . 4 0
4 snssi 3499 . . . 4 (0 ℝ → {0} ⊆ ℝ)
53, 4ax-mp 7 . . 3 {0} ⊆ ℝ
62, 5unssi 3112 . 2 (ℕ ∪ {0}) ⊆ ℝ
71, 6eqsstri 2969 1 0 ⊆ ℝ
Colors of variables: wff set class
Syntax hints:   wcel 1390  cun 2909  wss 2911  {csn 3367  cr 6710  0cc0 6711  cn 7695  0cn0 7957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-cnex 6774  ax-resscn 6775  ax-1re 6777  ax-addrcl 6780  ax-rnegex 6792
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-int 3607  df-inn 7696  df-n0 7958
This theorem is referenced by:  nn0sscn  7962  nn0re  7966  nn0rei  7968  nn0red  8012
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