Theorem ressxr 8087

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 3327	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 8082	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3219	1 |- RR C_ RR*

Syntax hints:   u. cun 3155   C_ wss 3157   {cpr 3624   RRcr 7895   +oocpnf 8075   -oocmnf 8076   RR*cxr 8077

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 8082

This theorem is referenced by:  rexpssxrxp  8088  rexr  8089  0xr  8090  rexrd  8093  ltrelxr  8104  iooval2  10007  fzval2  10103  seq3coll  10951  summodclem2a  11563  prodmodclem2a  11758  ismet2  14674  qtopbas  14842  tgqioo  14875