Theorem ressxr 8213

Description: The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)

Assertion

Ref	Expression
ressxr	⊢ ℝ ⊆ ℝ*

Proof of Theorem ressxr

Step	Hyp	Ref	Expression
1		ssun1 3368	. 2 ⊢ ℝ ⊆ (ℝ ∪ {+∞, -∞})
2		df-xr 8208	. 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞})
3	1, 2	sseqtrri 3260	1 ⊢ ℝ ⊆ ℝ*

Syntax hints:   ∪ cun 3196   ⊆ wss 3198  {cpr 3668  ℝcr 8021  +∞cpnf 8201  -∞cmnf 8202  ℝ^*cxr 8203

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 2802  df-un 3202  df-in 3204  df-ss 3211  df-xr 8208

This theorem is referenced by:  rexpssxrxp  8214  rexr  8215  0xr  8216  rexrd  8219  ltrelxr  8230  iooval2  10140  fzval2  10236  seq3coll  11096  summodclem2a  11932  prodmodclem2a  12127  ismet2  15068  qtopbas  15236  tgqioo  15269