Theorem elrp 9890

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 4092	. 2 |- ( x = A -> ( 0 < x <-> 0 < A ) )
2		df-rp 9889	. 2 |- RR+ = { x e. RR | 0 < x }
3	1, 2	elrab2 2965	1 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2202   class class class wbr 4088   RRcr 8031   0cc0 8032   < clt 8214   RR+crp 9888

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-rp 9889

This theorem is referenced by:  elrpii  9891  nnrp  9898  rpgt0  9900  rpregt0  9902  ralrp  9910  rexrp  9911  rpaddcl  9912  rpmulcl  9913  rpdivcl  9914  rpgecl  9917  rphalflt  9918  ge0p1rp  9920  rpnegap  9921  negelrp  9922  ltsubrp  9925  ltaddrp  9926  difrp  9927  elrpd  9928  iccdil  10233  icccntr  10235  dfrp2  10523  expgt0  10834  sqrtdiv  11603  mulcn2  11873  ef01bndlem  12318  nconstwlpolem  16672