Theorem nnssre 8925

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 7958	. 2 |- 1 e. RR
2		peano2re 8095	. . 3 |- ( x e. RR -> ( x + 1 ) e. RR )
3	2	rgen 2530	. 2 |- A. x e. RR ( x + 1 ) e. RR
4		peano5nni 8924	. 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 2148   A.wral 2455   C_ wss 3131  (class class class)co 5877   RRcr 7812   1c1 7814   + caddc 7816   NNcn 8921

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4123  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-in 3137  df-ss 3144  df-int 3847  df-inn 8922

This theorem is referenced by:  nnsscn  8926  nnre  8928  nnred  8934  nn0ssre  9182  nninfdclemp1  12453  nninfdclemf1  12455