Theorem prnmax 9274

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 2526	. . . . 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 4402	. . . . 5 |- ( y = B -> ( y <Q x <-> B <Q x ) )
4	3	rexbidv 2864	. . . 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 9267	. . . . . 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 2918	. . . 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 3134	. 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 1368   = wceq 1370   e. wcel 1758   A.wral 2798   E.wrex 2799   _Vcvv 3076   C. wpss 3436   (/)c0 3744   class class class wbr 4399   Q.cnq 9129   <Q cltq 9135   P.cnp 9136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-np 9260

This theorem is referenced by:  npomex  9275  prnmadd  9276  genpnmax  9286  1idpr  9308  ltexprlem4  9318  reclem3pr  9328  suplem1pr  9331