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Theorem elnp 8790
Description: Membership in positive reals. (Contributed by NM, 16-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
elnp  |-  ( A  e.  P.  <->  ( ( (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  ( A. y ( y  <Q  x  ->  y  e.  A
)  /\  E. y  e.  A  x  <Q  y ) ) )
Distinct variable group:    x, y, A

Proof of Theorem elnp
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elex 2900 . 2  |-  ( A  e.  P.  ->  A  e.  _V )
2 pssss 3378 . . . 4  |-  ( A 
C.  Q.  ->  A  C_  Q. )
3 nqex 8726 . . . . 5  |-  Q.  e.  _V
43ssex 4281 . . . 4  |-  ( A 
C_  Q.  ->  A  e. 
_V )
52, 4syl 16 . . 3  |-  ( A 
C.  Q.  ->  A  e. 
_V )
65ad2antlr 708 . 2  |-  ( ( ( (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  ( A. y ( y 
<Q  x  ->  y  e.  A )  /\  E. y  e.  A  x  <Q  y ) )  ->  A  e.  _V )
7 psseq2 3371 . . . . 5  |-  ( z  =  A  ->  ( (/)  C.  z  <->  (/)  C.  A )
)
8 psseq1 3370 . . . . 5  |-  ( z  =  A  ->  (
z  C.  Q.  <->  A  C.  Q. ) )
97, 8anbi12d 692 . . . 4  |-  ( z  =  A  ->  (
( (/)  C.  z  /\  z  C.  Q. )  <->  ( (/)  C.  A  /\  A  C.  Q. )
) )
10 eleq2 2441 . . . . . . . 8  |-  ( z  =  A  ->  (
y  e.  z  <->  y  e.  A ) )
1110imbi2d 308 . . . . . . 7  |-  ( z  =  A  ->  (
( y  <Q  x  ->  y  e.  z )  <-> 
( y  <Q  x  ->  y  e.  A ) ) )
1211albidv 1632 . . . . . 6  |-  ( z  =  A  ->  ( A. y ( y  <Q  x  ->  y  e.  z )  <->  A. y ( y 
<Q  x  ->  y  e.  A ) ) )
13 rexeq 2841 . . . . . 6  |-  ( z  =  A  ->  ( E. y  e.  z  x  <Q  y  <->  E. y  e.  A  x  <Q  y ) )
1412, 13anbi12d 692 . . . . 5  |-  ( z  =  A  ->  (
( A. y ( y  <Q  x  ->  y  e.  z )  /\  E. y  e.  z  x 
<Q  y )  <->  ( A. y ( y  <Q  x  ->  y  e.  A
)  /\  E. y  e.  A  x  <Q  y ) ) )
1514raleqbi1dv 2848 . . . 4  |-  ( z  =  A  ->  ( A. x  e.  z 
( A. y ( y  <Q  x  ->  y  e.  z )  /\  E. y  e.  z  x 
<Q  y )  <->  A. x  e.  A  ( A. y ( y  <Q  x  ->  y  e.  A
)  /\  E. y  e.  A  x  <Q  y ) ) )
169, 15anbi12d 692 . . 3  |-  ( z  =  A  ->  (
( ( (/)  C.  z  /\  z  C.  Q. )  /\  A. x  e.  z  ( A. y ( y  <Q  x  ->  y  e.  z )  /\  E. y  e.  z  x 
<Q  y ) )  <->  ( ( (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  ( A. y ( y  <Q  x  ->  y  e.  A
)  /\  E. y  e.  A  x  <Q  y ) ) ) )
17 df-np 8784 . . 3  |-  P.  =  { z  |  ( ( (/)  C.  z  /\  z  C.  Q. )  /\  A. x  e.  z  ( A. y ( y 
<Q  x  ->  y  e.  z )  /\  E. y  e.  z  x  <Q  y ) ) }
1816, 17elab2g 3020 . 2  |-  ( A  e.  _V  ->  ( A  e.  P.  <->  ( ( (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  ( A. y ( y  <Q  x  ->  y  e.  A
)  /\  E. y  e.  A  x  <Q  y ) ) ) )
191, 6, 18pm5.21nii 343 1  |-  ( A  e.  P.  <->  ( ( (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  ( A. y ( y  <Q  x  ->  y  e.  A
)  /\  E. y  e.  A  x  <Q  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   A.wral 2642   E.wrex 2643   _Vcvv 2892    C_ wss 3256    C. wpss 3257   (/)c0 3564   class class class wbr 4146   Q.cnq 8653    <Q cltq 8659   P.cnp 8660
This theorem is referenced by:  genpcl  8811  nqpr  8817  ltexprlem5  8843  reclem2pr  8851  suplem1pr  8855
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-ni 8675  df-nq 8715  df-np 8784
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