Theorem nn0ssxnn0 9043

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 3239	. 2 |- NN0 C_ ( NN0 u. { +oo } )
2		df-xnn0 9041	. 2 |- NN0* = ( NN0 u. { +oo } )
3	1, 2	sseqtrri 3132	1 |- NN0 C_ NN0*

Syntax hints:   u. cun 3069   C_ wss 3071   {csn 3527   +oocpnf 7797   NN0cn0 8977  NN0*cxnn0 9040

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-xnn0 9041

This theorem is referenced by:  nn0xnn0  9044  0xnn0  9046  nn0xnn0d  9049