Theorem ressxr 8370

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 3392	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 8365	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3283	1 |- RR C_ RR*

This proof depends on syntax axioms:   u. cun 3218   C_ wss 3220   {cpr 3710   RRcr 8179   +oocpnf 8358   -oocmnf 8359   RR*cxr 8360

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-xr 8365

This theorem is used by:  rexpssxrxp  8371  rexr  8372  0xr  8373  rexrd  8376  ltrelxr  8387  iooval2  10328  fzval2  10425  seq3coll  11310  summodclem2a  12167  prodmodclem2a  12362  ismet2  15546  qtopbas  15714  tgqioo  15747