Theorem ressxr 7833

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 3244	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 7828	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3137	1 |- RR C_ RR*

Syntax hints:   u. cun 3074   C_ wss 3076   {cpr 3533   RRcr 7643   +oocpnf 7821   -oocmnf 7822   RR*cxr 7823

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-xr 7828

This theorem is referenced by:  rexpssxrxp  7834  rexr  7835  0xr  7836  rexrd  7839  ltrelxr  7849  iooval2  9728  fzval2  9824  seq3coll  10617  summodclem2a  11182  prodmodclem2a  11377  ismet2  12562  qtopbas  12730  tgqioo  12755