Theorem rexri 8143

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 8131	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ*

Syntax hints:   ∈ wcel 2177  ℝcr 7937  ℝ^*cxr 8119

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3172  df-in 3174  df-ss 3181  df-xr 8124

This theorem is referenced by:  1xr  8144  cos12dec  12129  halfleoddlt  12255  reeff1oleme  15294  reeff1o  15295  sin0pilem2  15304  neghalfpirx  15316  sincosq1sgn  15348  sincosq2sgn  15349  sincosq4sgn  15351  sinq12gt0  15352  cosq14gt0  15354  cosq23lt0  15355  coseq0q4123  15356  coseq00topi  15357  coseq0negpitopi  15358  cosordlem  15371  cosq34lt1  15372  cos02pilt1  15373  cos0pilt1  15374  ioocosf1o  15376  negpitopissre  15377  iooref1o  16088  taupi  16127