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Theorem elnp 8627
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 2809 . 2  |-  ( A  e.  P.  ->  A  e.  _V )
2 pssss 3284 . . . 4  |-  ( A 
C.  Q.  ->  A  C_  Q. )
3 nqex 8563 . . . . 5  |-  Q.  e.  _V
43ssex 4174 . . . 4  |-  ( A 
C_  Q.  ->  A  e. 
_V )
52, 4syl 15 . . 3  |-  ( A 
C.  Q.  ->  A  e. 
_V )
65ad2antlr 707 . 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 3277 . . . . 5  |-  ( z  =  A  ->  ( (/)  C.  z  <->  (/)  C.  A )
)
8 psseq1 3276 . . . . 5  |-  ( z  =  A  ->  (
z  C.  Q.  <->  A  C.  Q. ) )
97, 8anbi12d 691 . . . 4  |-  ( z  =  A  ->  (
( (/)  C.  z  /\  z  C.  Q. )  <->  ( (/)  C.  A  /\  A  C.  Q. )
) )
10 eleq2 2357 . . . . . . . 8  |-  ( z  =  A  ->  (
y  e.  z  <->  y  e.  A ) )
1110imbi2d 307 . . . . . . 7  |-  ( z  =  A  ->  (
( y  <Q  x  ->  y  e.  z )  <-> 
( y  <Q  x  ->  y  e.  A ) ) )
1211albidv 1615 . . . . . 6  |-  ( z  =  A  ->  ( A. y ( y  <Q  x  ->  y  e.  z )  <->  A. y ( y 
<Q  x  ->  y  e.  A ) ) )
13 rexeq 2750 . . . . . 6  |-  ( z  =  A  ->  ( E. y  e.  z  x  <Q  y  <->  E. y  e.  A  x  <Q  y ) )
1412, 13anbi12d 691 . . . . 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 2757 . . . 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 691 . . 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 8621 . . 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 2929 . 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 342 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 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165    C. wpss 3166   (/)c0 3468   class class class wbr 4039   Q.cnq 8490    <Q cltq 8496   P.cnp 8497
This theorem is referenced by:  genpcl  8648  nqpr  8654  ltexprlem5  8680  reclem2pr  8688  suplem1pr  8692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-ni 8512  df-nq 8552  df-np 8621
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