Theorem rexri 8280

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 8268	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ*

Syntax hints:   ∈ wcel 2202  ℝcr 8074  ℝ^*cxr 8256

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-xr 8261

This theorem is referenced by:  1xr  8281  cos12dec  12390  halfleoddlt  12516  reeff1oleme  15563  reeff1o  15564  sin0pilem2  15573  neghalfpirx  15585  sincosq1sgn  15617  sincosq2sgn  15618  sincosq4sgn  15620  sinq12gt0  15621  cosq14gt0  15623  cosq23lt0  15624  coseq0q4123  15625  coseq00topi  15626  coseq0negpitopi  15627  cosordlem  15640  cosq34lt1  15641  cos02pilt1  15642  cos0pilt1  15643  ioocosf1o  15645  negpitopissre  15646  iooref1o  16746  taupi  16786