Theorem ressxr 8223

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 3370	. 2 ⊢ ℝ ⊆ (ℝ ∪ {+∞, -∞})
2		df-xr 8218	. 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞})
3	1, 2	sseqtrri 3262	1 ⊢ ℝ ⊆ ℝ*

Syntax hints:   ∪ cun 3198   ⊆ wss 3200  {cpr 3670  ℝcr 8031  +∞cpnf 8211  -∞cmnf 8212  ℝ^*cxr 8213

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-xr 8218

This theorem is referenced by:  rexpssxrxp  8224  rexr  8225  0xr  8226  rexrd  8229  ltrelxr  8240  iooval2  10150  fzval2  10246  seq3coll  11107  summodclem2a  11944  prodmodclem2a  12139  ismet2  15081  qtopbas  15249  tgqioo  15282