Theorem prn0 8826

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 8825	. . 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 3629	. 2 |- ( (/) C. A <-> A =/= (/) )
5	3, 4	sylib 189	1 |- ( A e. P. -> A =/= (/) )

Syntax hints:   -> wi 4   /\ wa 359   /\ w3a 936   A.wal 1546   e. wcel 1721   =/= wne 2571   A.wral 2670   E.wrex 2671   _Vcvv 2920   C. wpss 3285   (/)c0 3592   class class class wbr 4176   Q.cnq 8687   <Q cltq 8693   P.cnp 8694

This theorem is referenced by:  0npr  8829  npomex  8833  genpn0  8840  prlem934  8870  ltaddpr  8871  prlem936  8884  reclem2pr  8885  suplem1pr  8889

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389

This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-v 2922  df-dif 3287  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-np 8818