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Theorem prn0 6875
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 6874 . . 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 959 . . 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 195 . 2 |- (A e. P. -> (/) C. A)
4 0pss 2956 . 2 |- ((/) C. A <-> A =/= (/))
53, 4sylib 196 1 |- (A e. P. -> A =/= (/))
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 377   /\ w3a 937  A.wal 1366   e. wcel 1451   =/= wne 2071  A.wral 2162  E.wrex 2163  _Vcvv 2358   C. wpss 2658  (/)c0 2921   class class class wbr 3373  Q.cnq 6735   <Q cltq 6741  P.cnp 6742
This theorem is referenced by:  0npr 6878  npomex 6882  genpn0 6889  prlem934 6919  ltaddpr 6920  prlem936 6933  reclem2pr 6934  suplem1pr 6938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1367  ax-6 1368  ax-7 1369  ax-gen 1370  ax-8 1453  ax-10 1454  ax-11 1455  ax-12 1456  ax-17 1465  ax-9 1480  ax-4 1486  ax-16 1664  ax-ext 1935
This theorem depends on definitions:  df-bi 185  df-or 378  df-an 379  df-3an 939  df-ex 1372  df-sb 1626  df-clab 1941  df-cleq 1946  df-clel 1949  df-ne 2073  df-ral 2166  df-rex 2167  df-v 2360  df-dif 2660  df-in 2664  df-ss 2666  df-pss 2668  df-nul 2922  df-np 6867
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