Theorem nnssre 9243

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 8275	. 2 |- 1 e. RR
2		peano2re 8411	. . 3 |- ( x e. RR -> ( x + 1 ) e. RR )
3	2	rgen 2597	. 2 |- A. x e. RR ( x + 1 ) e. RR
4		peano5nni 9242	. 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 2205   A.wral 2522   C_ wss 3213  (class class class)co 6052   RRcr 8128   1c1 8130   + caddc 8132   NNcn 9239

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 2216  ax-sep 4230  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-in 3219  df-ss 3226  df-int 3952  df-inn 9240

This theorem is referenced by:  nnsscn  9244  nnre  9246  nnred  9252  nn0ssre  9502  nninfdclemp1  13218  nninfdclemf1  13220