Theorem ressxr 8063

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 3322	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 8058	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3214	1 |- RR C_ RR*

Syntax hints:   u. cun 3151   C_ wss 3153   {cpr 3619   RRcr 7871   +oocpnf 8051   -oocmnf 8052   RR*cxr 8053

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-xr 8058

This theorem is referenced by:  rexpssxrxp  8064  rexr  8065  0xr  8066  rexrd  8069  ltrelxr  8080  iooval2  9981  fzval2  10077  seq3coll  10913  summodclem2a  11524  prodmodclem2a  11719  ismet2  14522  qtopbas  14690  tgqioo  14715