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Theorem prnmax 8832
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 2468 . . . . 5  |-  ( y  =  B  ->  (
y  e.  A  <->  B  e.  A ) )
21anbi2d 685 . . . 4  |-  ( y  =  B  ->  (
( A  e.  P.  /\  y  e.  A )  <-> 
( A  e.  P.  /\  B  e.  A ) ) )
3 breq1 4179 . . . . 5  |-  ( y  =  B  ->  (
y  <Q  x  <->  B  <Q  x ) )
43rexbidv 2691 . . . 4  |-  ( y  =  B  ->  ( E. x  e.  A  y  <Q  x  <->  E. x  e.  A  B  <Q  x ) )
52, 4imbi12d 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 8825 . . . . . 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 451 . . . . 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 2768 . . . 4  |-  ( ( A  e.  P.  /\  y  e.  A )  ->  ( A. x ( x  <Q  y  ->  x  e.  A )  /\  E. x  e.  A  y 
<Q  x ) )
98simprd 450 . . 3  |-  ( ( A  e.  P.  /\  y  e.  A )  ->  E. x  e.  A  y  <Q  x )
105, 9vtoclg 2975 . 2  |-  ( B  e.  A  ->  (
( A  e.  P.  /\  B  e.  A )  ->  E. x  e.  A  B  <Q  x ) )
1110anabsi7 793 1  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  E. x  e.  A  B  <Q  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671   _Vcvv 2920    C. wpss 3285   (/)c0 3592   class class class wbr 4176   Q.cnq 8687    <Q cltq 8693   P.cnp 8694
This theorem is referenced by:  npomex  8833  prnmadd  8834  genpnmax  8844  1idpr  8866  ltexprlem4  8876  reclem3pr  8886  suplem1pr  8889
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 2389
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 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-np 8818
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