Theorem nn0ssxnn0 9180

Description: The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)

Assertion

Ref	Expression
nn0ssxnn0	|- NN0 C_ NN0*

Proof of Theorem nn0ssxnn0

Step	Hyp	Ref	Expression
1		ssun1 3285	. 2 |- NN0 C_ ( NN0 u. { +oo } )
2		df-xnn0 9178	. 2 |- NN0* = ( NN0 u. { +oo } )
3	1, 2	sseqtrri 3177	1 |- NN0 C_ NN0*

Syntax hints:   u. cun 3114   C_ wss 3116   {csn 3576   +oocpnf 7930   NN0cn0 9114  NN0*cxnn0 9177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-xnn0 9178

This theorem is referenced by:  nn0xnn0  9181  0xnn0  9183  nn0xnn0d  9186