Theorem nn0ssxnn0 9361

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 3336	. 2 |- NN0 C_ ( NN0 u. { +oo } )
2		df-xnn0 9359	. 2 |- NN0* = ( NN0 u. { +oo } )
3	1, 2	sseqtrri 3228	1 |- NN0 C_ NN0*

Syntax hints:   u. cun 3164   C_ wss 3166   {csn 3633   +oocpnf 8104   NN0cn0 9295  NN0*cxnn0 9358

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-xnn0 9359

This theorem is referenced by:  nn0xnn0  9362  0xnn0  9364  nn0xnn0d  9367  nninfctlemfo  12361