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Theorem prn0 6867
Description: A positive real is not empty. (Revised by Mario Carneiro, 11-May-2013.)
Assertion
Ref Expression
prn0 |- (A e. P. -> A =/= (/))

Proof of Theorem prn0
StepHypRef Expression
1 elnpi 6866 . . 3 |- (A e. P. <-> ((A e. _V /\ (/) C. A /\ A C. Q.) /\ A.x e. A (A.y(y <Q x -> y e. A) /\ E.y e. A x <Q y)))
2 simpl2 943 . . 3 |- (((A e. _V /\ (/) C. A /\ A C. Q.) /\ A.x e. A (A.y(y <Q x -> y e. A) /\ E.y e. A x <Q y)) -> (/) C. A)
31, 2sylbi 185 . 2 |- (A e. P. -> (/) C. A)
4 0pss 2941 . 2 |- ((/) C. A <-> A =/= (/))
53, 4sylib 186 1 |- (A e. P. -> A =/= (/))
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 361   /\ w3a 921  A.wal 1350   e. wcel 1436   =/= wne 2056  A.wral 2147  E.wrex 2148  _Vcvv 2343   C. wpss 2643  (/)c0 2906   class class class wbr 3362  Q.cnq 6727   <Q cltq 6733  P.cnp 6734
This theorem is referenced by:  0npr 6870  npomex 6874  genpn0 6881  prlem934 6911  ltaddpr 6912  prlem936 6925  reclem2pr 6926  suplem1pr 6930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1351  ax-6 1352  ax-7 1353  ax-gen 1354  ax-8 1438  ax-10 1439  ax-11 1440  ax-12 1441  ax-17 1450  ax-9 1465  ax-4 1471  ax-16 1649  ax-ext 1920
This theorem depends on definitions:  df-bi 175  df-or 362  df-an 363  df-3an 923  df-ex 1356  df-sb 1611  df-clab 1926  df-cleq 1931  df-clel 1934  df-ne 2058  df-ral 2151  df-rex 2152  df-v 2345  df-dif 2645  df-in 2649  df-ss 2651  df-pss 2653  df-nul 2907  df-np 6859
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