Theorem rexri 8084

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 8072	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ*

Syntax hints:   ∈ wcel 2167  ℝcr 7878  ℝ^*cxr 8060

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-xr 8065

This theorem is referenced by:  1xr  8085  cos12dec  11933  halfleoddlt  12059  reeff1oleme  15008  reeff1o  15009  sin0pilem2  15018  neghalfpirx  15030  sincosq1sgn  15062  sincosq2sgn  15063  sincosq4sgn  15065  sinq12gt0  15066  cosq14gt0  15068  cosq23lt0  15069  coseq0q4123  15070  coseq00topi  15071  coseq0negpitopi  15072  cosordlem  15085  cosq34lt1  15086  cos02pilt1  15087  cos0pilt1  15088  ioocosf1o  15090  negpitopissre  15091  iooref1o  15678  taupi  15717