Theorem nnssnn0 9516

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 3386	. 2 |- NN C_ ( NN u. { 0 } )
2		df-n0 9514	. 2 |- NN0 = ( NN u. { 0 } )
3	1, 2	sseqtrri 3277	1 |- NN C_ NN0

Syntax hints:   u. cun 3212   C_ wss 3214   {csn 3694   0cc0 8143   NNcn 9254   NN0cn0 9513

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-n0 9514

This theorem is referenced by:  nnnn0  9520  nnnn0d  9570  expcnv  12215  oddge22np1  12592  bitsfzolem  12665