Theorem elrp 9721

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 4033	. 2 |- ( x = A -> ( 0 < x <-> 0 < A ) )
2		df-rp 9720	. 2 |- RR+ = { x e. RR | 0 < x }
3	1, 2	elrab2 2919	1 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2164   class class class wbr 4029   RRcr 7871   0cc0 7872   < clt 8054   RR+crp 9719

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 3157  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-rp 9720

This theorem is referenced by:  elrpii  9722  nnrp  9729  rpgt0  9731  rpregt0  9733  ralrp  9741  rexrp  9742  rpaddcl  9743  rpmulcl  9744  rpdivcl  9745  rpgecl  9748  rphalflt  9749  ge0p1rp  9751  rpnegap  9752  negelrp  9753  ltsubrp  9756  ltaddrp  9757  difrp  9758  elrpd  9759  iccdil  10064  icccntr  10066  dfrp2  10332  expgt0  10643  sqrtdiv  11186  mulcn2  11455  ef01bndlem  11899  nconstwlpolem  15555