Theorem elrp 9724

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 4034	. 2 |- ( x = A -> ( 0 < x <-> 0 < A ) )
2		df-rp 9723	. 2 |- RR+ = { x e. RR | 0 < x }
3	1, 2	elrab2 2920	1 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2164   class class class wbr 4030   RRcr 7873   0cc0 7874   < clt 8056   RR+crp 9722

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-rp 9723

This theorem is referenced by:  elrpii  9725  nnrp  9732  rpgt0  9734  rpregt0  9736  ralrp  9744  rexrp  9745  rpaddcl  9746  rpmulcl  9747  rpdivcl  9748  rpgecl  9751  rphalflt  9752  ge0p1rp  9754  rpnegap  9755  negelrp  9756  ltsubrp  9759  ltaddrp  9760  difrp  9761  elrpd  9762  iccdil  10067  icccntr  10069  dfrp2  10335  expgt0  10646  sqrtdiv  11189  mulcn2  11458  ef01bndlem  11902  nconstwlpolem  15625