Theorem nnssre 9075

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 8106	. 2 |- 1 e. RR
2		peano2re 8243	. . 3 |- ( x e. RR -> ( x + 1 ) e. RR )
3	2	rgen 2561	. 2 |- A. x e. RR ( x + 1 ) e. RR
4		peano5nni 9074	. 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 2178   A.wral 2486   C_ wss 3174  (class class class)co 5967   RRcr 7959   1c1 7961   + caddc 7963   NNcn 9071

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-sep 4178  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-in 3180  df-ss 3187  df-int 3900  df-inn 9072

This theorem is referenced by:  nnsscn  9076  nnre  9078  nnred  9084  nn0ssre  9334  nninfdclemp1  12936  nninfdclemf1  12938