Theorem ressxr 7510

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 3161	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 7505	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtr4i 3057	1 |- RR C_ RR*

Syntax hints:   u. cun 2995   C_ wss 2997   {cpr 3442   RRcr 7328   +oocpnf 7498   -oocmnf 7499   RR*cxr 7500

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-xr 7505

This theorem is referenced by:  rexpssxrxp  7511  rexr  7512  0xr  7513  rexrd  7516  ltrelxr  7526  iooval2  9302  fzval2  9396  iseqcoll  10212  isummolem2a  10735