Theorem ressxr 7260

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 3146	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 7255	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtr4i 3042	1 |- RR C_ RR*

Syntax hints:   u. cun 2981   C_ wss 2983   {cpr 3418   RRcr 7078   +oocpnf 7248   -oocmnf 7249   RR*cxr 7250

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-xr 7255

This theorem is referenced by:  rexpssxrxp  7261  rexr  7262  0xr  7263  rexrd  7266  ltrelxr  7276  iooval2  9050  fzval2  9144