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Theorem rpre 8364
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 8359 . . 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 6710  0cc0 6711   < clt 6857  +crp 8358
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-bndl 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 8359
This theorem is referenced by:  rpxr  8365  rpcn  8366  rpssre  8368  rpge0  8370  rprege0  8372  rpap0  8374  rprene0  8375  rpreap0  8376  rpaddcl  8381  rpmulcl  8382  rpdivcl  8383  rpgecl  8386  iccdil  8636  expnlbnd  9026  rennim  9211
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