Theorem nnssre 9011

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 8042	. 2 |- 1 e. RR
2		peano2re 8179	. . 3 |- ( x e. RR -> ( x + 1 ) e. RR )
3	2	rgen 2550	. 2 |- A. x e. RR ( x + 1 ) e. RR
4		peano5nni 9010	. 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 2167   A.wral 2475   C_ wss 3157  (class class class)co 5925   RRcr 7895   1c1 7897   + caddc 7899   NNcn 9007

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-sep 4152  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-int 3876  df-inn 9008

This theorem is referenced by:  nnsscn  9012  nnre  9014  nnred  9020  nn0ssre  9270  nninfdclemp1  12692  nninfdclemf1  12694