Theorem nn0ssxnn0 8491

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 3145	. 2 |- NN0 C_ ( NN0 u. { +oo } )
2		df-xnn0 8489	. 2 |- NN0* = ( NN0 u. { +oo } )
3	1, 2	sseqtr4i 3041	1 |- NN0 C_ NN0*

Syntax hints:   u. cun 2980   C_ wss 2982   {csn 3416   +oocpnf 7282   NN0cn0 8425  NN0*cxnn0 8488

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-xnn0 8489

This theorem is referenced by:  nn0xnn0  8492  0xnn0  8494  nn0xnn0d  8497