Theorem ressxr 7069

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 3106	. 2 |- RR C_ ( RR u. { +oo , -oo } )
2		df-xr 7064	. 2 |- RR* = ( RR u. { +oo , -oo } )
3	1, 2	sseqtr4i 2978	1 |- RR C_ RR*

Syntax hints:   u. cun 2915   C_ wss 2917   {cpr 3376   RRcr 6888   +oocpnf 7057   -oocmnf 7058   RR*cxr 7059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-xr 7064

This theorem is referenced by:  rexpssxrxp  7070  rexr  7071  0xr  7072  rexrd  7075  ltrelxr  7080  iooval2  8784  fzval2  8877