Theorem nnssnn0 8774

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 3178	. 2 |- NN C_ ( NN u. { 0 } )
2		df-n0 8772	. 2 |- NN0 = ( NN u. { 0 } )
3	1, 2	sseqtr4i 3074	1 |- NN C_ NN0

Syntax hints:   u. cun 3011   C_ wss 3013   {csn 3466   0cc0 7447   NNcn 8520   NN0cn0 8771

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-n0 8772

This theorem is referenced by:  nnnn0  8778  nnnn0d  8824  expcnv  11062  oddge22np1  11323