Theorem ressxr 8036

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 3313	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 8031	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3205	1 |- RR C_ RR*

Syntax hints:   u. cun 3142   C_ wss 3144   {cpr 3611   RRcr 7845   +oocpnf 8024   -oocmnf 8025   RR*cxr 8026

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 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-xr 8031

This theorem is referenced by:  rexpssxrxp  8037  rexr  8038  0xr  8039  rexrd  8042  ltrelxr  8053  iooval2  9951  fzval2  10047  seq3coll  10863  summodclem2a  11430  prodmodclem2a  11625  ismet2  14339  qtopbas  14507  tgqioo  14532