Theorem nn0ssxnn0 9431

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 3367	. 2 |- NN0 C_ ( NN0 u. { +oo } )
2		df-xnn0 9429	. 2 |- NN0* = ( NN0 u. { +oo } )
3	1, 2	sseqtrri 3259	1 |- NN0 C_ NN0*

Syntax hints:   u. cun 3195   C_ wss 3197   {csn 3666   +oocpnf 8174   NN0cn0 9365  NN0*cxnn0 9428

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-xnn0 9429

This theorem is referenced by:  nn0xnn0  9432  0xnn0  9434  nn0xnn0d  9437  nninfctlemfo  12556