Theorem nn0ssre 9373

Description: Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)

Assertion

Ref	Expression
nn0ssre	|- NN0 C_ RR

Proof of Theorem nn0ssre

Step	Hyp	Ref	Expression
1		df-n0 9370	. 2 |- NN0 = ( NN u. { 0 } )
2		nnssre 9114	. . 3 |- NN C_ RR
3		0re 8146	. . . 4 |- 0 e. RR
4		snssi 3812	. . . 4 |- ( 0 e. RR -> { 0 } C_ RR )
5	3, 4	ax-mp 5	. . 3 |- { 0 } C_ RR
6	2, 5	unssi 3379	. 2 |- ( NN u. { 0 } ) C_ RR
7	1, 6	eqsstri 3256	1 |- NN0 C_ RR

Syntax hints:   e. wcel 2200   u. cun 3195   C_ wss 3197   {csn 3666   RRcr 7998   0cc0 7999   NNcn 9110   NN0cn0 9369

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-sep 4202  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096  ax-rnegex 8108

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-int 3924  df-inn 9111  df-n0 9370

This theorem is referenced by:  nn0sscn  9374  nn0re  9378  nn0rei  9380  nn0red  9423