Theorem ressxr 7802

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 3234	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 7797	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3127	1 |- RR C_ RR*

Syntax hints:   u. cun 3064   C_ wss 3066   {cpr 3523   RRcr 7612   +oocpnf 7790   -oocmnf 7791   RR*cxr 7792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-xr 7797

This theorem is referenced by:  rexpssxrxp  7803  rexr  7804  0xr  7805  rexrd  7808  ltrelxr  7818  iooval2  9691  fzval2  9786  seq3coll  10578  summodclem2a  11143  ismet2  12512  qtopbas  12680  tgqioo  12705