Theorem nnssre 8882

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 7919	. 2 |- 1 e. RR
2		peano2re 8055	. . 3 |- ( x e. RR -> ( x + 1 ) e. RR )
3	2	rgen 2523	. 2 |- A. x e. RR ( x + 1 ) e. RR
4		peano5nni 8881	. 2 |- ( ( 1 e. RR /\ A. x e. RR ( x + 1 ) e. RR ) -> NN C_ RR )
5	1, 3, 4	mp2an 424	1 |- NN C_ RR

Syntax hints:   e. wcel 2141   A.wral 2448   C_ wss 3121  (class class class)co 5853   RRcr 7773   1c1 7775   + caddc 7777   NNcn 8878

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-int 3832  df-inn 8879

This theorem is referenced by:  nnsscn  8883  nnre  8885  nnred  8891  nn0ssre  9139  nninfdclemp1  12405  nninfdclemf1  12407