Theorem ressxr 7162

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 3135	. 2 ⊢ ℝ ⊆ (ℝ ∪ {+∞, -∞})
2		df-xr 7157	. 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞})
3	1, 2	sseqtr4i 3032	1 ⊢ ℝ ⊆ ℝ*

Syntax hints:   ∪ cun 2971   ⊆ wss 2973  {cpr 3399  ℝcr 6980  +∞cpnf 7150  -∞cmnf 7151  ℝ^*cxr 7152

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-xr 7157

This theorem is referenced by:  rexpssxrxp  7163  rexr  7164  0xr  7165  rexrd  7168  ltrelxr  7173  iooval2  8938  fzval2  9032