Theorem elrp 8869

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 3809	. 2 |- ( x = A -> ( 0 < x <-> 0 < A ) )
2		df-rp 8868	. 2 |- RR+ = { x e. RR | 0 < x }
3	1, 2	elrab2 2760	1 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )

Syntax hints:   /\ wa 102   <-> wb 103   e. wcel 1434   class class class wbr 3805   RRcr 7094   0cc0 7095   < clt 7267   RR+crp 8867

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-rp 8868

This theorem is referenced by:  elrpii  8870  nnrp  8876  rpgt0  8878  rpregt0  8880  ralrp  8888  rexrp  8889  rpaddcl  8890  rpmulcl  8891  rpdivcl  8892  rpgecl  8895  rphalflt  8896  ge0p1rp  8898  rpnegap  8899  ltsubrp  8901  ltaddrp  8902  difrp  8903  elrpd  8904  iccdil  9148  icccntr  9150  expgt0  9658  sqrtdiv  10129  mulcn2  10352