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Theorem prnmax 8709
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 2418 . . . . 5  |-  ( y  =  B  ->  (
y  e.  A  <->  B  e.  A ) )
21anbi2d 684 . . . 4  |-  ( y  =  B  ->  (
( A  e.  P.  /\  y  e.  A )  <-> 
( A  e.  P.  /\  B  e.  A ) ) )
3 breq1 4107 . . . . 5  |-  ( y  =  B  ->  (
y  <Q  x  <->  B  <Q  x ) )
43rexbidv 2640 . . . 4  |-  ( y  =  B  ->  ( E. x  e.  A  y  <Q  x  <->  E. x  e.  A  B  <Q  x ) )
52, 4imbi12d 311 . . 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 8702 . . . . . 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 450 . . . . 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 2717 . . . 4  |-  ( ( A  e.  P.  /\  y  e.  A )  ->  ( A. x ( x  <Q  y  ->  x  e.  A )  /\  E. x  e.  A  y 
<Q  x ) )
98simprd 449 . . 3  |-  ( ( A  e.  P.  /\  y  e.  A )  ->  E. x  e.  A  y  <Q  x )
105, 9vtoclg 2919 . 2  |-  ( B  e.  A  ->  (
( A  e.  P.  /\  B  e.  A )  ->  E. x  e.  A  B  <Q  x ) )
1110anabsi7 792 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 358    /\ w3a 934   A.wal 1540    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620   _Vcvv 2864    C. wpss 3229   (/)c0 3531   class class class wbr 4104   Q.cnq 8564    <Q cltq 8570   P.cnp 8571
This theorem is referenced by:  npomex  8710  prnmadd  8711  genpnmax  8721  1idpr  8743  ltexprlem4  8753  reclem3pr  8763  suplem1pr  8766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-np 8695
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