Theorem nn0ssre 9448

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 9445	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
2		nnssre 9189	. . 3 ⊢ ℕ ⊆ ℝ
3		0re 8222	. . . 4 ⊢ 0 ∈ ℝ
4		snssi 3822	. . . 4 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
5	3, 4	ax-mp 5	. . 3 ⊢ {0} ⊆ ℝ
6	2, 5	unssi 3384	. 2 ⊢ (ℕ ∪ {0}) ⊆ ℝ
7	1, 6	eqsstri 3260	1 ⊢ ℕ0 ⊆ ℝ

Syntax hints:   ∈ wcel 2202   ∪ cun 3199   ⊆ wss 3201  {csn 3673  ℝcr 8074  0cc0 8075  ℕcn 9185  ℕ₀cn0 9444

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-sep 4212  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172  ax-rnegex 8184

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-int 3934  df-inn 9186  df-n0 9445

This theorem is referenced by:  nn0sscn  9449  nn0re  9453  nn0rei  9455  nn0red  9500