Theorem rexri 8237

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 8225	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ*

Syntax hints:   ∈ wcel 2202  ℝcr 8031  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:  1xr  8238  cos12dec  12334  halfleoddlt  12460  reeff1oleme  15502  reeff1o  15503  sin0pilem2  15512  neghalfpirx  15524  sincosq1sgn  15556  sincosq2sgn  15557  sincosq4sgn  15559  sinq12gt0  15560  cosq14gt0  15562  cosq23lt0  15563  coseq0q4123  15564  coseq00topi  15565  coseq0negpitopi  15566  cosordlem  15579  cosq34lt1  15580  cos02pilt1  15581  cos0pilt1  15582  ioocosf1o  15584  negpitopissre  15585  iooref1o  16664  taupi  16703