Theorem nn0ssxnn0 9315

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 3326	. 2 |- NN0 C_ ( NN0 u. { +oo } )
2		df-xnn0 9313	. 2 |- NN0* = ( NN0 u. { +oo } )
3	1, 2	sseqtrri 3218	1 |- NN0 C_ NN0*

Syntax hints:   u. cun 3155   C_ wss 3157   {csn 3622   +oocpnf 8058   NN0cn0 9249  NN0*cxnn0 9312

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-xnn0 9313

This theorem is referenced by:  nn0xnn0  9316  0xnn0  9318  nn0xnn0d  9321  nninfctlemfo  12207