Theorem nnssre 9125

Description: The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)

Assertion

Ref	Expression
nnssre	⊢ ℕ ⊆ ℝ

Proof of Theorem nnssre

Step	Hyp	Ref	Expression
1		1re 8156	. 2 ⊢ 1 ∈ ℝ
2		peano2re 8293	. . 3 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
3	2	rgen 2583	. 2 ⊢ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ
4		peano5nni 9124	. 2 ⊢ ((1 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ) → ℕ ⊆ ℝ)
5	1, 3, 4	mp2an 426	1 ⊢ ℕ ⊆ ℝ

Syntax hints:   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197  (class class class)co 6007  ℝcr 8009  1c1 8011   + caddc 8013  ℕcn 9121

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 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

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-v 2801  df-in 3203  df-ss 3210  df-int 3924  df-inn 9122

This theorem is referenced by:  nnsscn  9126  nnre  9128  nnred  9134  nn0ssre  9384  nninfdclemp1  13036  nninfdclemf1  13038