Theorem nn0ssxnn0 9201

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 3290	. 2 |- NN0 C_ ( NN0 u. { +oo } )
2		df-xnn0 9199	. 2 |- NN0* = ( NN0 u. { +oo } )
3	1, 2	sseqtrri 3182	1 |- NN0 C_ NN0*

Syntax hints:   u. cun 3119   C_ wss 3121   {csn 3583   +oocpnf 7951   NN0cn0 9135  NN0*cxnn0 9198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-xnn0 9199

This theorem is referenced by:  nn0xnn0  9202  0xnn0  9204  nn0xnn0d  9207