Theorem elrp 6220

Description: Membership in the set of positive reals.

Assertion

Ref	Expression
elrp	|- (A e. RR+ <-> (A e. RR /\ 0 < A))

Proof of Theorem elrp

Step	Hyp	Ref	Expression
1		breq2 2613	. 2 |- (x = A -> (0 < x <-> 0 < A))
2		df-rp 6219	. 2 |- RR+ = {x e. RR | 0 < x}
3	1, 2	elrab2 1898	1 |- (A e. RR+ <-> (A e. RR /\ 0 < A))

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 955   class class class wbr 2609  RRcr 5205  0cc0 5206  RR+crp 5272   < clt 5458

This theorem is referenced by:  elrpi 6221  rpgt0t 6224  ralrp 6226  rpaddclt 6227  rpmulclt 6228  rpdivclt 6229  0nrp 6230  rpsqrclt 6645  absrpclt 6790  clmi2rp 7026  mulc1cncf 7214  ivthlem2 7217  ivthlem6OLD 7230  ivthlem7OLD 7231  efcn 7363  cncfmet 7844  lmcvgnns 7879  effoi 8666  effoiOLD 8667  dmse1 10467  ltsubpostb 10471  ltaddpos2tb 10472  iintlem1 10476  iintlem2 10477

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-rab 1644  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-rp 6219

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