Theorem nnssnn0 9333

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 3344	. 2 |- NN C_ ( NN u. { 0 } )
2		df-n0 9331	. 2 |- NN0 = ( NN u. { 0 } )
3	1, 2	sseqtrri 3236	1 |- NN C_ NN0

Syntax hints:   u. cun 3172   C_ wss 3174   {csn 3643   0cc0 7960   NNcn 9071   NN0cn0 9330

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

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-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-n0 9331

This theorem is referenced by:  nnnn0  9337  nnnn0d  9383  expcnv  11930  oddge22np1  12307  bitsfzolem  12380