Theorem nn0rei 9380

Description: A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)

Hypothesis

Ref	Expression
nn0re.1	⊢ 𝐴 ∈ ℕ0

Assertion

Ref	Expression
nn0rei	⊢ 𝐴 ∈ ℝ

Proof of Theorem nn0rei

Step	Hyp	Ref	Expression
1		nn0ssre 9373	. 2 ⊢ ℕ0 ⊆ ℝ
2		nn0re.1	. 2 ⊢ 𝐴 ∈ ℕ0
3	1, 2	sselii 3221	1 ⊢ 𝐴 ∈ ℝ

Syntax hints:   ∈ wcel 2200  ℝcr 7998  ℕ₀cn0 9369

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-sep 4202  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096  ax-rnegex 8108

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-int 3924  df-inn 9111  df-n0 9370

This theorem is referenced by:  nn0cni  9381  nn0le2xi  9419  nn0lele2xi  9420  numlt  9602  numltc  9603  decle  9611  decleh  9612  modsubi  12942