Theorem rexri 8377

Description: A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)

Hypothesis

Ref	Expression
rexri.1	⊢ 𝐴 ∈ ℝ

Assertion

Ref	Expression
rexri	⊢ 𝐴 ∈ ℝ*

Proof of Theorem rexri

Step	Hyp	Ref	Expression
1		rexri.1	. 2 ⊢ 𝐴 ∈ ℝ
2		rexr 8365	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ*

Syntax hints:   ∈ wcel 2209  ℝcr 8172  ℝ^*cxr 8353

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 8358

This theorem is referenced by:  1xr  8378  cos12dec  12518  halfleoddlt  12644  reeff1oleme  15856  reeff1o  15857  sin0pilem2  15866  neghalfpirx  15878  sincosq1sgn  15910  sincosq2sgn  15911  sincosq4sgn  15913  sinq12gt0  15914  cosq14gt0  15916  cosq23lt0  15917  coseq0q4123  15918  coseq00topi  15919  coseq0negpitopi  15920  cosordlem  15933  cosq34lt1  15934  cos02pilt1  15935  cos0pilt1  15936  ioocosf1o  15938  negpitopissre  15939  iooref1o  17057  taupi  17097