Theorem prn0 8581

Description: A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
prn0	|- ( A e. P. -> A =/= (/) )

Proof of Theorem prn0

Step	Hyp	Ref	Expression
1		elnpi 8580	. . 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 964	. . 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 189	. 2 |- ( A e. P. -> (/) C. A )
4		0pss 3467	. 2 |- ( (/) C. A <-> A =/= (/) )
5	3, 4	sylib 190	1 |- ( A e. P. -> A =/= (/) )

Syntax hints:   -> wi 6   /\ wa 360   /\ w3a 939   A.wal 1532   e. wcel 1621   =/= wne 2421   A.wral 2518   E.wrex 2519   _Vcvv 2763   C. wpss 3128   (/)c0 3430   class class class wbr 3997   Q.cnq 8442   <Q cltq 8448   P.cnp 8449

This theorem is referenced by:  0npr  8584  npomex  8588  genpn0  8595  prlem934  8625  ltaddpr  8626  prlem936  8639  reclem2pr  8640  suplem1pr  8644

This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-v 2765  df-dif 3130  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-np 8573