Theorem ressxr 8333

Description: The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)

Assertion

Ref	Expression
ressxr	|- RR C_ RR*

Proof of Theorem ressxr

Step	Hyp	Ref	Expression
1		ssun1 3386	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 8328	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3277	1 |- RR C_ RR*

Syntax hints:   u. cun 3212   C_ wss 3214   {cpr 3695   RRcr 8142   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-xr 8328

This theorem is referenced by:  rexpssxrxp  8334  rexr  8335  0xr  8336  rexrd  8339  ltrelxr  8350  iooval2  10267  fzval2  10364  seq3coll  11239  summodclem2a  12092  prodmodclem2a  12287  ismet2  15345  qtopbas  15513  tgqioo  15546