Theorem rexri 8347

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 8335	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ*

Syntax hints:   ∈ wcel 2205  ℝcr 8142  ℝ^*cxr 8323

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-xr 8328

This theorem is referenced by:  1xr  8348  cos12dec  12479  halfleoddlt  12605  reeff1oleme  15749  reeff1o  15750  sin0pilem2  15759  neghalfpirx  15771  sincosq1sgn  15803  sincosq2sgn  15804  sincosq4sgn  15806  sinq12gt0  15807  cosq14gt0  15809  cosq23lt0  15810  coseq0q4123  15811  coseq00topi  15812  coseq0negpitopi  15813  cosordlem  15826  cosq34lt1  15827  cos02pilt1  15828  cos0pilt1  15829  ioocosf1o  15831  negpitopissre  15832  iooref1o  16930  taupi  16971