Theorem ressxr 8070

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 3326	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 8065	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3218	1 |- RR C_ RR*

Syntax hints:   u. cun 3155   C_ wss 3157   {cpr 3623   RRcr 7878   +oocpnf 8058   -oocmnf 8059   RR*cxr 8060

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-xr 8065

This theorem is referenced by:  rexpssxrxp  8071  rexr  8072  0xr  8073  rexrd  8076  ltrelxr  8087  iooval2  9990  fzval2  10086  seq3coll  10934  summodclem2a  11546  prodmodclem2a  11741  ismet2  14590  qtopbas  14758  tgqioo  14791