Theorem nn0ssxnn0 9238

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 3298	. 2 |- NN0 C_ ( NN0 u. { +oo } )
2		df-xnn0 9236	. 2 |- NN0* = ( NN0 u. { +oo } )
3	1, 2	sseqtrri 3190	1 |- NN0 C_ NN0*

Syntax hints:   u. cun 3127   C_ wss 3129   {csn 3592   +oocpnf 7985   NN0cn0 9172  NN0*cxnn0 9235

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-xnn0 9236

This theorem is referenced by:  nn0xnn0  9239  0xnn0  9241  nn0xnn0d  9244