Theorem ressxr 8359

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 3392	. 2 ⊢ ℝ ⊆ (ℝ ∪ {+∞, -∞})
2		df-xr 8354	. 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞})
3	1, 2	sseqtrri 3283	1 ⊢ ℝ ⊆ ℝ*

Syntax hints:   ∪ cun 3218   ⊆ wss 3220  {cpr 3706  ℝcr 8168  +∞cpnf 8347  -∞cmnf 8348  ℝ^*cxr 8349

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-xr 8354

This theorem is referenced by:  rexpssxrxp  8360  rexr  8361  0xr  8362  rexrd  8365  ltrelxr  8376  iooval2  10296  fzval2  10393  seq3coll  11272  summodclem2a  12126  prodmodclem2a  12321  ismet2  15378  qtopbas  15546  tgqioo  15579