Theorem rexri 8230

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 8218	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ*

Syntax hints:   ∈ wcel 2200  ℝcr 8024  ℝ^*cxr 8206

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 2802  df-un 3202  df-in 3204  df-ss 3211  df-xr 8211

This theorem is referenced by:  1xr  8231  cos12dec  12322  halfleoddlt  12448  reeff1oleme  15489  reeff1o  15490  sin0pilem2  15499  neghalfpirx  15511  sincosq1sgn  15543  sincosq2sgn  15544  sincosq4sgn  15546  sinq12gt0  15547  cosq14gt0  15549  cosq23lt0  15550  coseq0q4123  15551  coseq00topi  15552  coseq0negpitopi  15553  cosordlem  15566  cosq34lt1  15567  cos02pilt1  15568  cos0pilt1  15569  ioocosf1o  15571  negpitopissre  15572  iooref1o  16588  taupi  16627