Theorem prn0 8858

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

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnpi 8857	. . 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 961	. . 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 188	. 2 |- ( A e. P. -> (/) C. A )
4		0pss 3657	. 2 |- ( (/) C. A <-> A =/= (/) )
5	3, 4	sylib 189	1 |- ( A e. P. -> A =/= (/) )

Syntax hints:   -> wi 4   /\ wa 359   /\ w3a 936   A.wal 1549   e. wcel 1725   =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948   C. wpss 3313   (/)c0 3620   class class class wbr 4204   Q.cnq 8719   <Q cltq 8725   P.cnp 8726

This theorem is referenced by:  0npr  8861  npomex  8865  genpn0  8872  prlem934  8902  ltaddpr  8903  prlem936  8916  reclem2pr  8917  suplem1pr  8921

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-np 8850