Theorem prn0 6897

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

Step	Hyp	Ref	Expression
1		elnpi 6896	. . 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 934	. . 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)
3	1, 2	sylbi 185	. 2 |- (A e. P. -> (/) C. A)
4		0pss 2940	. 2 |- ((/) C. A <-> A =/= (/))
5	3, 4	sylib 186	1 |- (A e. P. -> A =/= (/))

Syntax hints:   -> wi 4   /\ wa 360   /\ w3a 912  A.wal 1341   e. wcel 1427   =/= wne 2047  A.wral 2138  E.wrex 2139  _Vcvv 2335   C. wpss 2639  (/)c0 2903   class class class wbr 3362  Q.cnq 6757   <Q cltq 6763  P.cnp 6764

This theorem is referenced by:  0npr 6900  npomex 6904  genpn0 6911  prlem934 6941  ltaddpr 6942  prlem936 6955  reclem2pr 6956  suplem1pr 6960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1342  ax-6 1343  ax-7 1344  ax-gen 1345  ax-8 1429  ax-10 1430  ax-11 1431  ax-12 1432  ax-17 1441  ax-9 1456  ax-4 1462  ax-16 1640  ax-ext 1911

This theorem depends on definitions:  df-bi 175  df-or 361  df-an 362  df-3an 914  df-ex 1347  df-sb 1602  df-clab 1917  df-cleq 1922  df-clel 1925  df-ne 2049  df-ral 2142  df-rex 2143  df-v 2337  df-dif 2641  df-in 2645  df-ss 2647  df-pss 2649  df-nul 2904  df-np 6889

Copyright terms: Public domain