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Theorem rpre 8344
Description: A positive real is a real. (Contributed by NM, 27-Oct-2007.)
Assertion
Ref Expression
rpre (A +A ℝ)

Proof of Theorem rpre
StepHypRef Expression
1 df-rp 8339 . . 3 + = {x ℝ ∣ 0 < x}
2 ssrab2 3019 . . 3 {x ℝ ∣ 0 < x} ⊆ ℝ
31, 2eqsstri 2969 . 2 + ⊆ ℝ
43sseli 2935 1 (A +A ℝ)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1390  {crab 2304   class class class wbr 3755  cr 6690  0cc0 6691   < clt 6837  +crp 8338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-in 2918  df-ss 2925  df-rp 8339
This theorem is referenced by:  rpxr  8345  rpcn  8346  rpssre  8348  rpge0  8350  rprege0  8352  rpap0  8354  rprene0  8355  rpreap0  8356  rpaddcl  8361  rpmulcl  8362  rpdivcl  8363  rpgecl  8366  iccdil  8616  expnlbnd  9006  rennim  9191
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