Theorem nnssnn0 9545

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 3392	. 2 |- NN C_ ( NN u. { 0 } )
2		df-n0 9543	. 2 |- NN0 = ( NN u. { 0 } )
3	1, 2	sseqtrri 3283	1 |- NN C_ NN0

Syntax hints:   u. cun 3218   C_ wss 3220   {csn 3705   0cc0 8169   NNcn 9283   NN0cn0 9542

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-n0 9543

This theorem is referenced by:  nnnn0  9549  nnnn0d  9599  expcnv  12249  oddge22np1  12626  bitsfzolem  12699