Theorem nn0ssre 9405

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 9402	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
2		nnssre 9146	. . 3 ⊢ ℕ ⊆ ℝ
3		0re 8178	. . . 4 ⊢ 0 ∈ ℝ
4		snssi 3817	. . . 4 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
5	3, 4	ax-mp 5	. . 3 ⊢ {0} ⊆ ℝ
6	2, 5	unssi 3382	. 2 ⊢ (ℕ ∪ {0}) ⊆ ℝ
7	1, 6	eqsstri 3259	1 ⊢ ℕ0 ⊆ ℝ

Syntax hints:   ∈ wcel 2202   ∪ cun 3198   ⊆ wss 3200  {csn 3669  ℝcr 8030  0cc0 8031  ℕcn 9142  ℕ₀cn0 9401

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-sep 4207  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128  ax-rnegex 8140

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-int 3929  df-inn 9143  df-n0 9402

This theorem is referenced by:  nn0sscn  9406  nn0re  9410  nn0rei  9412  nn0red  9455