Theorem nnssre 8857

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 7894	. 2 |- 1 e. RR
2		peano2re 8030	. . 3 |- ( x e. RR -> ( x + 1 ) e. RR )
3	2	rgen 2518	. 2 |- A. x e. RR ( x + 1 ) e. RR
4		peano5nni 8856	. 2 |- ( ( 1 e. RR /\ A. x e. RR ( x + 1 ) e. RR ) -> NN C_ RR )
5	1, 3, 4	mp2an 423	1 |- NN C_ RR

Syntax hints:   e. wcel 2136   A.wral 2443   C_ wss 3115  (class class class)co 5841   RRcr 7748   1c1 7750   + caddc 7752   NNcn 8853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4099  ax-cnex 7840  ax-resscn 7841  ax-1re 7843  ax-addrcl 7846

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-v 2727  df-in 3121  df-ss 3128  df-int 3824  df-inn 8854

This theorem is referenced by:  nnsscn  8858  nnre  8860  nnred  8866  nn0ssre  9114  nninfdclemp1  12379  nninfdclemf1  12381