Theorem prnmax 9371

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 2494	. . . . 5 |- ( y = B -> ( y e. A <-> B e. A ) )
2	1	anbi2d 708	. . . 4 |- ( y = B -> ( ( A e. P. /\ y e. A ) <-> ( A e. P. /\ B e. A ) ) )
3		breq1 4369	. . . . 5 |- ( y = B -> ( y <Q x <-> B <Q x ) )
4	3	rexbidv 2878	. . . 4 |- ( y = B -> ( E. x e. A y <Q x <-> E. x e. A B <Q x ) )
5	2, 4	imbi12d 321	. . 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 9364	. . . . . 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 465	. . . . 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 2734	. . . 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 464	. . 3 |- ( ( A e. P. /\ y e. A ) -> E. x e. A y <Q x )
10	5, 9	vtoclg 3082	. 2 |- ( B e. A -> ( ( A e. P. /\ B e. A ) -> E. x e. A B <Q x ) )
11	10	anabsi7 826	1 |- ( ( A e. P. /\ B e. A ) -> E. x e. A B <Q x )

Syntax hints:   -> wi 4   /\ wa 370   /\ w3a 982   A.wal 1435   = wceq 1437   e. wcel 1872   A.wral 2714   E.wrex 2715   _Vcvv 3022   C. wpss 3380   (/)c0 3704   class class class wbr 4366   Q.cnq 9228   <Q cltq 9234   P.cnp 9235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-np 9357

This theorem is referenced by:  npomex  9372  prnmadd  9373  genpnmax  9383  1idpr  9405  ltexprlem4  9415  reclem3pr  9425  suplem1pr  9428