Theorem elrp 9730

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 4037	. 2 |- ( x = A -> ( 0 < x <-> 0 < A ) )
2		df-rp 9729	. 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 4033   RRcr 7878   0cc0 7879   < clt 8061   RR+crp 9728

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 3628  df-pr 3629  df-op 3631  df-br 4034  df-rp 9729

This theorem is referenced by:  elrpii  9731  nnrp  9738  rpgt0  9740  rpregt0  9742  ralrp  9750  rexrp  9751  rpaddcl  9752  rpmulcl  9753  rpdivcl  9754  rpgecl  9757  rphalflt  9758  ge0p1rp  9760  rpnegap  9761  negelrp  9762  ltsubrp  9765  ltaddrp  9766  difrp  9767  elrpd  9768  iccdil  10073  icccntr  10075  dfrp2  10353  expgt0  10664  sqrtdiv  11207  mulcn2  11477  ef01bndlem  11921  nconstwlpolem  15709