Theorem ressxr 8190

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 3367	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 8185	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3259	1 |- RR C_ RR*

Syntax hints:   u. cun 3195   C_ wss 3197   {cpr 3667   RRcr 7998   +oocpnf 8178   -oocmnf 8179   RR*cxr 8180

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-xr 8185

This theorem is referenced by:  rexpssxrxp  8191  rexr  8192  0xr  8193  rexrd  8196  ltrelxr  8207  iooval2  10111  fzval2  10207  seq3coll  11064  summodclem2a  11892  prodmodclem2a  12087  ismet2  15028  qtopbas  15196  tgqioo  15229