Theorem rexri 8220

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 8208	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ*

Syntax hints:   ∈ wcel 2200  ℝcr 8014  ℝ^*cxr 8196

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 8201

This theorem is referenced by:  1xr  8221  cos12dec  12300  halfleoddlt  12426  reeff1oleme  15467  reeff1o  15468  sin0pilem2  15477  neghalfpirx  15489  sincosq1sgn  15521  sincosq2sgn  15522  sincosq4sgn  15524  sinq12gt0  15525  cosq14gt0  15527  cosq23lt0  15528  coseq0q4123  15529  coseq00topi  15530  coseq0negpitopi  15531  cosordlem  15544  cosq34lt1  15545  cos02pilt1  15546  cos0pilt1  15547  ioocosf1o  15549  negpitopissre  15550  iooref1o  16516  taupi  16555