Theorem elrp 9747

Description: Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)

Assertion

Ref	Expression
elrp	|- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )

Proof of Theorem elrp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4038	. 2 |- ( x = A -> ( 0 < x <-> 0 < A ) )
2		df-rp 9746	. 2 |- RR+ = { x e. RR | 0 < x }
3	1, 2	elrab2 2923	1 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2167   class class class wbr 4034   RRcr 7895   0cc0 7896   < clt 8078   RR+crp 9745

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-rp 9746

This theorem is referenced by:  elrpii  9748  nnrp  9755  rpgt0  9757  rpregt0  9759  ralrp  9767  rexrp  9768  rpaddcl  9769  rpmulcl  9770  rpdivcl  9771  rpgecl  9774  rphalflt  9775  ge0p1rp  9777  rpnegap  9778  negelrp  9779  ltsubrp  9782  ltaddrp  9783  difrp  9784  elrpd  9785  iccdil  10090  icccntr  10092  dfrp2  10370  expgt0  10681  sqrtdiv  11224  mulcn2  11494  ef01bndlem  11938  nconstwlpolem  15796