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Theorem elrp 8683
Description: Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
Assertion
Ref Expression
elrp (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))

Proof of Theorem elrp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 breq2 3796 . 2 (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴))
2 df-rp 8682 . 2 + = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
31, 2elrab2 2723 1 (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wcel 1409   class class class wbr 3792  cr 6946  0cc0 6947   < clt 7119  +crp 8681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-rp 8682
This theorem is referenced by:  elrpii  8684  nnrp  8690  rpgt0  8692  rpregt0  8694  ralrp  8702  rexrp  8703  rpaddcl  8704  rpmulcl  8705  rpdivcl  8706  rpgecl  8709  rphalflt  8710  ge0p1rp  8712  rpnegap  8713  ltsubrp  8715  ltaddrp  8716  difrp  8717  elrpd  8718  iccdil  8967  icccntr  8969  expgt0  9453  sqrtdiv  9869  mulcn2  10064
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