Theorem ressxr 8003

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 3300	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 7998	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtrri 3192	1 |- RR C_ RR*

Syntax hints:   u. cun 3129   C_ wss 3131   {cpr 3595   RRcr 7812   +oocpnf 7991   -oocmnf 7992   RR*cxr 7993

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-xr 7998

This theorem is referenced by:  rexpssxrxp  8004  rexr  8005  0xr  8006  rexrd  8009  ltrelxr  8020  iooval2  9917  fzval2  10013  seq3coll  10824  summodclem2a  11391  prodmodclem2a  11586  ismet2  13893  qtopbas  14061  tgqioo  14086