Theorem elrp 8736

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 3789	. 2 |- ( x = A -> ( 0 < x <-> 0 < A ) )
2		df-rp 8735	. 2 |- RR+ = { x e. RR | 0 < x }
3	1, 2	elrab2 2751	1 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )

Syntax hints:   /\ wa 102   <-> wb 103   e. wcel 1433   class class class wbr 3785   RRcr 6980   0cc0 6981   < clt 7153   RR+crp 8734

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-rp 8735

This theorem is referenced by:  elrpii  8737  nnrp  8743  rpgt0  8745  rpregt0  8747  ralrp  8755  rexrp  8756  rpaddcl  8757  rpmulcl  8758  rpdivcl  8759  rpgecl  8762  rphalflt  8763  ge0p1rp  8765  rpnegap  8766  ltsubrp  8768  ltaddrp  8769  difrp  8770  elrpd  8771  iccdil  9020  icccntr  9022  expgt0  9509  sqrtdiv  9928  mulcn2  10151