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Theorem rpgt0 9878
Description: A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
Assertion
Ref Expression
rpgt0 (𝐴 ∈ ℝ+ → 0 < 𝐴)

Proof of Theorem rpgt0
StepHypRef Expression
1 elrp 9868 . 2 (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
21simprbi 275 1 (𝐴 ∈ ℝ+ → 0 < 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2200   class class class wbr 4083  cr 8014  0cc0 8015   < clt 8197  +crp 9866
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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-rp 9867
This theorem is referenced by:  rpge0  9879  rpap0  9883  rpgecl  9895  0nrp  9902  rpgt0d  9912  addlelt  9981  rpsqrtcl  11573  rpmaxcl  11755  rpmincl  11770  xrminrpcl  11806  climconst  11822  sinltxirr  12293  blcntrps  15110  blcntr  15111  bdmet  15197  bdmopn  15199  reeff1o  15468  coseq00topi  15530  coseq0negpitopi  15531
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