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Theorem prnmax 8573
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
StepHypRef Expression
1 eleq1 2316 . . . . 5  |-  ( y  =  B  ->  (
y  e.  A  <->  B  e.  A ) )
21anbi2d 687 . . . 4  |-  ( y  =  B  ->  (
( A  e.  P.  /\  y  e.  A )  <-> 
( A  e.  P.  /\  B  e.  A ) ) )
3 breq1 3986 . . . . 5  |-  ( y  =  B  ->  (
y  <Q  x  <->  B  <Q  x ) )
43rexbidv 2537 . . . 4  |-  ( y  =  B  ->  ( E. x  e.  A  y  <Q  x  <->  E. x  e.  A  B  <Q  x ) )
52, 4imbi12d 313 . . 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 8566 . . . . . 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 452 . . . . 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 2614 . . . 4  |-  ( ( A  e.  P.  /\  y  e.  A )  ->  ( A. x ( x  <Q  y  ->  x  e.  A )  /\  E. x  e.  A  y 
<Q  x ) )
98simprd 451 . . 3  |-  ( ( A  e.  P.  /\  y  e.  A )  ->  E. x  e.  A  y  <Q  x )
105, 9vtoclg 2811 . 2  |-  ( B  e.  A  ->  (
( A  e.  P.  /\  B  e.  A )  ->  E. x  e.  A  B  <Q  x ) )
1110anabsi7 795 1  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  E. x  e.  A  B  <Q  x )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939   A.wal 1532    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517   _Vcvv 2757    C. wpss 3114   (/)c0 3416   class class class wbr 3983   Q.cnq 8428    <Q cltq 8434   P.cnp 8435
This theorem is referenced by:  npomex  8574  prnmadd  8575  genpnmax  8585  1idpr  8607  ltexprlem4  8617  reclem3pr  8627  suplem1pr  8630
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-br 3984  df-np 8559
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