Theorem nn0ssxnn0 9309

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 3323	. 2 |- NN0 C_ ( NN0 u. { +oo } )
2		df-xnn0 9307	. 2 |- NN0* = ( NN0 u. { +oo } )
3	1, 2	sseqtrri 3215	1 |- NN0 C_ NN0*

Syntax hints:   u. cun 3152   C_ wss 3154   {csn 3619   +oocpnf 8053   NN0cn0 9243  NN0*cxnn0 9306

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-xnn0 9307

This theorem is referenced by:  nn0xnn0  9310  0xnn0  9312  nn0xnn0d  9315  nninfctlemfo  12180