Theorem nnssre 9311

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 8326	. 2 |- 1 e. RR
2		peano2re 8464	. . 3 |- ( x e. RR -> ( x + 1 ) e. RR )
3	2	rgen 2603	. 2 |- A. x e. RR ( x + 1 ) e. RR
4		peano5nni 9310	. 2 |- ( ( 1 e. RR /\ A. x e. RR ( x + 1 ) e. RR ) -> NN C_ RR )
5	1, 3, 4	mp2an 430	1 |- NN C_ RR

This proof depends on syntax axioms:   e. wcel 2209   A.wral 2528   C_ wss 3220  (class class class)co 6085   RRcr 8179   1c1 8181   + caddc 8183   NNcn 9307

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4249  ax-cnex 8271  ax-resscn 8272  ax-1re 8274  ax-addrcl 8277

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-in 3226  df-ss 3233  df-int 3971  df-inn 9308

This theorem is used by:  nnsscn  9312  nnre  9314  nnred  9320  nn0ssre  9572  nninfdclemp1  13393  nninfdclemf1  13395