Theorem prnmax 8828

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 2464	. . . . 5 |- ( y = B -> ( y e. A <-> B e. A ) )
2	1	anbi2d 685	. . . 4 |- ( y = B -> ( ( A e. P. /\ y e. A ) <-> ( A e. P. /\ B e. A ) ) )
3		breq1 4175	. . . . 5 |- ( y = B -> ( y <Q x <-> B <Q x ) )
4	3	rexbidv 2687	. . . 4 |- ( y = B -> ( E. x e. A y <Q x <-> E. x e. A B <Q x ) )
5	2, 4	imbi12d 312	. . 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 8821	. . . . . 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 451	. . . . 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 2764	. . . 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 450	. . 3 |- ( ( A e. P. /\ y e. A ) -> E. x e. A y <Q x )
10	5, 9	vtoclg 2971	. 2 |- ( B e. A -> ( ( A e. P. /\ B e. A ) -> E. x e. A B <Q x ) )
11	10	anabsi7 793	1 |- ( ( A e. P. /\ B e. A ) -> E. x e. A B <Q x )

Syntax hints:   -> wi 4   /\ wa 359   /\ w3a 936   A.wal 1546   = wceq 1649   e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916   C. wpss 3281   (/)c0 3588   class class class wbr 4172   Q.cnq 8683   <Q cltq 8689   P.cnp 8690

This theorem is referenced by:  npomex  8829  prnmadd  8830  genpnmax  8840  1idpr  8862  ltexprlem4  8872  reclem3pr  8882  suplem1pr  8885

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 2385

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-np 8814