Theorem nnssnn0 9447

Description: Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Assertion

Ref	Expression
nnssnn0	|- NN C_ NN0

Proof of Theorem nnssnn0

Step	Hyp	Ref	Expression
1		ssun1 3372	. 2 |- NN C_ ( NN u. { 0 } )
2		df-n0 9445	. 2 |- NN0 = ( NN u. { 0 } )
3	1, 2	sseqtrri 3263	1 |- NN C_ NN0

Syntax hints:   u. cun 3199   C_ wss 3201   {csn 3673   0cc0 8075   NNcn 9185   NN0cn0 9444

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-n0 9445

This theorem is referenced by:  nnnn0  9451  nnnn0d  9499  expcnv  12128  oddge22np1  12505  bitsfzolem  12578