Theorem prn0 9811

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 9810	. . 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 1065	. . 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 207	. 2 |- ( A e. P. -> (/) C. A )
4		0pss 4013	. 2 |- ( (/) C. A <-> A =/= (/) )
5	3, 4	sylib 208	1 |- ( A e. P. -> A =/= (/) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C. wpss 3575   (/)c0 3915   class class class wbr 4653   Q.cnq 9674   <Q cltq 9680   P.cnp 9681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-np 9803

This theorem is referenced by:  0npr  9814  npomex  9818  genpn0  9825  prlem934  9855  ltaddpr  9856  prlem936  9869  reclem2pr  9870  suplem1pr  9874