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Theorem elnp 8856
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 2956 . 2  |-  ( A  e.  P.  ->  A  e.  _V )
2 pssss 3434 . . . 4  |-  ( A 
C.  Q.  ->  A  C_  Q. )
3 nqex 8792 . . . . 5  |-  Q.  e.  _V
43ssex 4339 . . . 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 3427 . . . . 5  |-  ( z  =  A  ->  ( (/)  C.  z  <->  (/)  C.  A )
)
8 psseq1 3426 . . . . 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 2496 . . . . . . . 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 1635 . . . . . 6  |-  ( z  =  A  ->  ( A. y ( y  <Q  x  ->  y  e.  z )  <->  A. y ( y 
<Q  x  ->  y  e.  A ) ) )
13 rexeq 2897 . . . . . 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 2904 . . . 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 8850 . . 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 3076 . 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 1549    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    C_ wss 3312    C. wpss 3313   (/)c0 3620   class class class wbr 4204   Q.cnq 8719    <Q cltq 8725   P.cnp 8726
This theorem is referenced by:  genpcl  8877  nqpr  8883  ltexprlem5  8909  reclem2pr  8917  suplem1pr  8921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-ni 8741  df-nq 8781  df-np 8850
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