Theorem nnssre 9237

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	|- NN C_ RR

Proof of Theorem nnssre

Step	Hyp	Ref	Expression
1		1re 8269	. 2 |- 1 e. RR
2		peano2re 8405	. . 3 |- ( x e. RR -> ( x + 1 ) e. RR )
3	2	rgen 2595	. 2 |- A. x e. RR ( x + 1 ) e. RR
4		peano5nni 9236	. 2 |- ( ( 1 e. RR /\ A. x e. RR ( x + 1 ) e. RR ) -> NN C_ RR )
5	1, 3, 4	mp2an 426	1 |- NN C_ RR

Syntax hints:   e. wcel 2203   A.wral 2520   C_ wss 3210  (class class class)co 6049   RRcr 8122   1c1 8124   + caddc 8126   NNcn 9233

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 2214  ax-sep 4227  ax-cnex 8214  ax-resscn 8215  ax-1re 8217  ax-addrcl 8220

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2814  df-in 3216  df-ss 3223  df-int 3949  df-inn 9234

This theorem is referenced by:  nnsscn  9238  nnre  9240  nnred  9246  nn0ssre  9496  nninfdclemp1  13190  nninfdclemf1  13192