Theorem rexri 8330

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 8318	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ*

Syntax hints:   ∈ wcel 2203  ℝcr 8125  ℝ^*cxr 8306

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-xr 8311

This theorem is referenced by:  1xr  8331  cos12dec  12450  halfleoddlt  12576  reeff1oleme  15629  reeff1o  15630  sin0pilem2  15639  neghalfpirx  15651  sincosq1sgn  15683  sincosq2sgn  15684  sincosq4sgn  15686  sinq12gt0  15687  cosq14gt0  15689  cosq23lt0  15690  coseq0q4123  15691  coseq00topi  15692  coseq0negpitopi  15693  cosordlem  15706  cosq34lt1  15707  cos02pilt1  15708  cos0pilt1  15709  ioocosf1o  15711  negpitopissre  15712  iooref1o  16810  taupi  16850