Theorem nn0ssre 9396

Description: Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)

Assertion

Ref	Expression
nn0ssre	⊢ ℕ0 ⊆ ℝ

Proof of Theorem nn0ssre

Step	Hyp	Ref	Expression
1		df-n0 9393	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
2		nnssre 9137	. . 3 ⊢ ℕ ⊆ ℝ
3		0re 8169	. . . 4 ⊢ 0 ∈ ℝ
4		snssi 3815	. . . 4 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
5	3, 4	ax-mp 5	. . 3 ⊢ {0} ⊆ ℝ
6	2, 5	unssi 3380	. 2 ⊢ (ℕ ∪ {0}) ⊆ ℝ
7	1, 6	eqsstri 3257	1 ⊢ ℕ0 ⊆ ℝ

Syntax hints:   ∈ wcel 2200   ∪ cun 3196   ⊆ wss 3198  {csn 3667  ℝcr 8021  0cc0 8022  ℕcn 9133  ℕ₀cn0 9392

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 4205  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119  ax-rnegex 8131

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 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-int 3927  df-inn 9134  df-n0 9393

This theorem is referenced by:  nn0sscn  9397  nn0re  9401  nn0rei  9403  nn0red  9446