Theorem ressxr 8116

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 3336	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 8111	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3228	1 |- RR C_ RR*

Syntax hints:   u. cun 3164   C_ wss 3166   {cpr 3634   RRcr 7924   +oocpnf 8104   -oocmnf 8105   RR*cxr 8106

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-xr 8111

This theorem is referenced by:  rexpssxrxp  8117  rexr  8118  0xr  8119  rexrd  8122  ltrelxr  8133  iooval2  10037  fzval2  10133  seq3coll  10987  summodclem2a  11692  prodmodclem2a  11887  ismet2  14826  qtopbas  14994  tgqioo  15027