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Theorem prnmax 9362
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 2526 . . . . 5  |-  ( y  =  B  ->  (
y  e.  A  <->  B  e.  A ) )
21anbi2d 701 . . . 4  |-  ( y  =  B  ->  (
( A  e.  P.  /\  y  e.  A )  <-> 
( A  e.  P.  /\  B  e.  A ) ) )
3 breq1 4442 . . . . 5  |-  ( y  =  B  ->  (
y  <Q  x  <->  B  <Q  x ) )
43rexbidv 2965 . . . 4  |-  ( y  =  B  ->  ( E. x  e.  A  y  <Q  x  <->  E. x  e.  A  B  <Q  x ) )
52, 4imbi12d 318 . . 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 9355 . . . . . 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 462 . . . . 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 2823 . . . 4  |-  ( ( A  e.  P.  /\  y  e.  A )  ->  ( A. x ( x  <Q  y  ->  x  e.  A )  /\  E. x  e.  A  y 
<Q  x ) )
98simprd 461 . . 3  |-  ( ( A  e.  P.  /\  y  e.  A )  ->  E. x  e.  A  y  <Q  x )
105, 9vtoclg 3164 . 2  |-  ( B  e.  A  ->  (
( A  e.  P.  /\  B  e.  A )  ->  E. x  e.  A  B  <Q  x ) )
1110anabsi7 817 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 367    /\ w3a 971   A.wal 1396    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106    C. wpss 3462   (/)c0 3783   class class class wbr 4439   Q.cnq 9219    <Q cltq 9225   P.cnp 9226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-np 9348
This theorem is referenced by:  npomex  9363  prnmadd  9364  genpnmax  9374  1idpr  9396  ltexprlem4  9406  reclem3pr  9416  suplem1pr  9419
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