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Theorem elrpii 12385
Description: Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
Hypotheses
Ref Expression
elrpi.1 𝐴 ∈ ℝ
elrpi.2 0 < 𝐴
Assertion
Ref Expression
elrpii 𝐴 ∈ ℝ+

Proof of Theorem elrpii
StepHypRef Expression
1 elrpi.1 . 2 𝐴 ∈ ℝ
2 elrpi.2 . 2 0 < 𝐴
3 elrp 12384 . 2 (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
41, 2, 3mpbir2an 707 1 𝐴 ∈ ℝ+
Colors of variables: wff setvar class
Syntax hints:  wcel 2107   class class class wbr 5062  cr 10528  0cc0 10529   < clt 10667  +crp 12382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-rp 12383
This theorem is referenced by:  1rp  12386  2rp  12387  3rp  12388  iexpcyc  13562  discr  13594  epr  15553  aaliou3lem1  24846  aaliou3lem2  24847  aaliou3lem3  24848  pirp  24962  pigt3  25018  efif1olem2  25040  cxpsqrtlem  25198  log2cnv  25436  chtublem  25701  chtub  25702  bposlem6  25779  lgsdir2lem1  25815  lgsdir2lem4  25818  lgsdir2lem5  25819  2sqlem11  25919  chebbnd1lem3  25961  chebbnd1  25962  pntlemg  26088  pntlemr  26092  pntlemf  26095  minvecolem3  28567  dp2lt10  30474  ballotlem2  31632  cntotbnd  34942  heiborlem5  34961  heiborlem7  34963  isosctrlem1ALT  41129  sineq0ALT  41132  limclner  41793  stoweidlem5  42152  stoweidlem28  42175  stoweidlem59  42206  stoweid  42210  stirlinglem12  42232  fourierswlem  42377  fouriersw  42378
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