ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elrp GIF version

Theorem elrp 9450
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 3933 . 2 (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴))
2 df-rp 9449 . 2 + = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
31, 2elrab2 2843 1 (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 1480   class class class wbr 3929  cr 7626  0cc0 7627   < clt 7807  +crp 9448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-rp 9449
This theorem is referenced by:  elrpii  9451  nnrp  9458  rpgt0  9460  rpregt0  9462  ralrp  9470  rexrp  9471  rpaddcl  9472  rpmulcl  9473  rpdivcl  9474  rpgecl  9477  rphalflt  9478  ge0p1rp  9480  rpnegap  9481  negelrp  9482  ltsubrp  9485  ltaddrp  9486  difrp  9487  elrpd  9488  iccdil  9788  icccntr  9790  expgt0  10333  sqrtdiv  10821  mulcn2  11088  ef01bndlem  11469
  Copyright terms: Public domain W3C validator