Theorem prnmax 9152

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 2493	. . . . 5 |- ( y = B -> ( y e. A <-> B e. A ) )
2	1	anbi2d 696	. . . 4 |- ( y = B -> ( ( A e. P. /\ y e. A ) <-> ( A e. P. /\ B e. A ) ) )
3		breq1 4283	. . . . 5 |- ( y = B -> ( y <Q x <-> B <Q x ) )
4	3	rexbidv 2726	. . . 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 9145	. . . . . 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 461	. . . . 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 2804	. . . 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 460	. . 3 |- ( ( A e. P. /\ y e. A ) -> E. x e. A y <Q x )
10	5, 9	vtoclg 3019	. 2 |- ( B e. A -> ( ( A e. P. /\ B e. A ) -> E. x e. A B <Q x ) )
11	10	anabsi7 808	1 |- ( ( A e. P. /\ B e. A ) -> E. x e. A B <Q x )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 958   A.wal 1360   = wceq 1362   e. wcel 1755   A.wral 2705   E.wrex 2706   _Vcvv 2962   C. wpss 3317   (/)c0 3625   class class class wbr 4280   Q.cnq 9007   <Q cltq 9013   P.cnp 9014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-np 9138

This theorem is referenced by:  npomex  9153  prnmadd  9154  genpnmax  9164  1idpr  9186  ltexprlem4  9196  reclem3pr  9206  suplem1pr  9209