Theorem prnmax 9156

Description: A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
prnmax	|- ( ( A e. P. /\ B e. A ) -> E. x e. A B <Q x )

Distinct variable groups:   x, A   x, B

Proof of Theorem prnmax

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2498	. . . . 5 |- ( y = B -> ( y e. A <-> B e. A ) )
2	1	anbi2d 703	. . . 4 |- ( y = B -> ( ( A e. P. /\ y e. A ) <-> ( A e. P. /\ B e. A ) ) )
3		breq1 4290	. . . . 5 |- ( y = B -> ( y <Q x <-> B <Q x ) )
4	3	rexbidv 2731	. . . 4 |- ( y = B -> ( E. x e. A y <Q x <-> E. x e. A B <Q x ) )
5	2, 4	imbi12d 320	. . 3 |- ( y = B -> ( ( ( A e. P. /\ y e. A ) -> E. x e. A y <Q x ) <-> ( ( A e. P. /\ B e. A ) -> E. x e. A B <Q x ) ) )
6		elnpi 9149	. . . . . 6 |- ( A e. P. <-> ( ( A e. _V /\ (/) C. A /\ A C. Q. ) /\ A. y e. A ( A. x ( x <Q y -> x e. A ) /\ E. x e. A y <Q x ) ) )
7	6	simprbi 464	. . . . 5 |- ( A e. P. -> A. y e. A ( A. x ( x <Q y -> x e. A ) /\ E. x e. A y <Q x ) )
8	7	r19.21bi 2809	. . . 4 |- ( ( A e. P. /\ y e. A ) -> ( A. x ( x <Q y -> x e. A ) /\ E. x e. A y <Q x ) )
9	8	simprd 463	. . 3 |- ( ( A e. P. /\ y e. A ) -> E. x e. A y <Q x )
10	5, 9	vtoclg 3025	. 2 |- ( B e. A -> ( ( A e. P. /\ B e. A ) -> E. x e. A B <Q x ) )
11	10	anabsi7 815	1 |- ( ( A e. P. /\ B e. A ) -> E. x e. A B <Q x )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   A.wal 1367   = wceq 1369   e. wcel 1756   A.wral 2710   E.wrex 2711   _Vcvv 2967   C. wpss 3324   (/)c0 3632   class class class wbr 4287   Q.cnq 9011   <Q cltq 9017   P.cnp 9018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-np 9142

This theorem is referenced by:  npomex  9157  prnmadd  9158  genpnmax  9168  1idpr  9190  ltexprlem4  9200  reclem3pr  9210  suplem1pr  9213