Theorem rexri 8192

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 8180	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ*

Syntax hints:   ∈ wcel 2200  ℝcr 7986  ℝ^*cxr 8168

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 8173

This theorem is referenced by:  1xr  8193  cos12dec  12265  halfleoddlt  12391  reeff1oleme  15431  reeff1o  15432  sin0pilem2  15441  neghalfpirx  15453  sincosq1sgn  15485  sincosq2sgn  15486  sincosq4sgn  15488  sinq12gt0  15489  cosq14gt0  15491  cosq23lt0  15492  coseq0q4123  15493  coseq00topi  15494  coseq0negpitopi  15495  cosordlem  15508  cosq34lt1  15509  cos02pilt1  15510  cos0pilt1  15511  ioocosf1o  15513  negpitopissre  15514  iooref1o  16333  taupi  16372