Theorem ressxr 7963

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 3290	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 7958	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3182	1 |- RR C_ RR*

Syntax hints:   u. cun 3119   C_ wss 3121   {cpr 3584   RRcr 7773   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-xr 7958

This theorem is referenced by:  rexpssxrxp  7964  rexr  7965  0xr  7966  rexrd  7969  ltrelxr  7980  iooval2  9872  fzval2  9968  seq3coll  10777  summodclem2a  11344  prodmodclem2a  11539  ismet2  13148  qtopbas  13316  tgqioo  13341