Theorem ressxr 8317

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 3382	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 8312	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3273	1 |- RR C_ RR*

Syntax hints:   u. cun 3209   C_ wss 3211   {cpr 3690   RRcr 8126   +oocpnf 8305   -oocmnf 8306   RR*cxr 8307

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-xr 8312

This theorem is referenced by:  rexpssxrxp  8318  rexr  8319  0xr  8320  rexrd  8323  ltrelxr  8334  iooval2  10248  fzval2  10345  seq3coll  11214  summodclem2a  12067  prodmodclem2a  12262  ismet2  15219  qtopbas  15387  tgqioo  15420