Theorem rexri 8212

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 8200	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ*

Syntax hints:   ∈ wcel 2200  ℝcr 8006  ℝ^*cxr 8188

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-xr 8193

This theorem is referenced by:  1xr  8213  cos12dec  12287  halfleoddlt  12413  reeff1oleme  15454  reeff1o  15455  sin0pilem2  15464  neghalfpirx  15476  sincosq1sgn  15508  sincosq2sgn  15509  sincosq4sgn  15511  sinq12gt0  15512  cosq14gt0  15514  cosq23lt0  15515  coseq0q4123  15516  coseq00topi  15517  coseq0negpitopi  15518  cosordlem  15531  cosq34lt1  15532  cos02pilt1  15533  cos0pilt1  15534  ioocosf1o  15536  negpitopissre  15537  iooref1o  16432  taupi  16471