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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.
StepHypRef Expression
1 eleq1 2493 . . . . 5  |-  ( y  =  B  ->  (
y  e.  A  <->  B  e.  A ) )
21anbi2d 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 ) )
43rexbidv 2726 . . . 4  |-  ( y  =  B  ->  ( E. x  e.  A  y  <Q  x  <->  E. x  e.  A  B  <Q  x ) )
52, 4imbi12d 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 ) ) )
76simprbi 461 . . . . 5  |-  ( A  e.  P.  ->  A. y  e.  A  ( A. x ( x  <Q  y  ->  x  e.  A
)  /\  E. x  e.  A  y  <Q  x ) )
87r19.21bi 2804 . . . 4  |-  ( ( A  e.  P.  /\  y  e.  A )  ->  ( A. x ( x  <Q  y  ->  x  e.  A )  /\  E. x  e.  A  y 
<Q  x ) )
98simprd 460 . . 3  |-  ( ( A  e.  P.  /\  y  e.  A )  ->  E. x  e.  A  y  <Q  x )
105, 9vtoclg 3019 . 2  |-  ( B  e.  A  ->  (
( A  e.  P.  /\  B  e.  A )  ->  E. x  e.  A  B  <Q  x ) )
1110anabsi7 808 1  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  E. x  e.  A  B  <Q  x )
Colors of variables: wff setvar class
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
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