MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elnp Structured version   Unicode version

Theorem elnp 9259
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 3079 . 2  |-  ( A  e.  P.  ->  A  e.  _V )
2 pssss 3551 . . . 4  |-  ( A 
C.  Q.  ->  A  C_  Q. )
3 nqex 9195 . . . . 5  |-  Q.  e.  _V
43ssex 4536 . . . 4  |-  ( A 
C_  Q.  ->  A  e. 
_V )
52, 4syl 16 . . 3  |-  ( A 
C.  Q.  ->  A  e. 
_V )
65ad2antlr 726 . 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 3544 . . . . 5  |-  ( z  =  A  ->  ( (/)  C.  z  <->  (/)  C.  A )
)
8 psseq1 3543 . . . . 5  |-  ( z  =  A  ->  (
z  C.  Q.  <->  A  C.  Q. )
)
97, 8anbi12d 710 . . . 4  |-  ( z  =  A  ->  (
( (/)  C.  z  /\  z  C.  Q. )  <->  ( (/)  C.  A  /\  A  C.  Q. )
) )
10 eleq2 2524 . . . . . . . 8  |-  ( z  =  A  ->  (
y  e.  z  <->  y  e.  A ) )
1110imbi2d 316 . . . . . . 7  |-  ( z  =  A  ->  (
( y  <Q  x  ->  y  e.  z )  <-> 
( y  <Q  x  ->  y  e.  A ) ) )
1211albidv 1680 . . . . . 6  |-  ( z  =  A  ->  ( A. y ( y  <Q  x  ->  y  e.  z )  <->  A. y ( y 
<Q  x  ->  y  e.  A ) ) )
13 rexeq 3016 . . . . . 6  |-  ( z  =  A  ->  ( E. y  e.  z  x  <Q  y  <->  E. y  e.  A  x  <Q  y ) )
1412, 13anbi12d 710 . . . . 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 3023 . . . 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 710 . . 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 9253 . . 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 3207 . 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 353 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   _Vcvv 3070    C_ wss 3428    C. wpss 3429   (/)c0 3737   class class class wbr 4392   Q.cnq 9122    <Q cltq 9128   P.cnp 9129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-tr 4486  df-eprel 4732  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-om 6579  df-ni 9144  df-nq 9184  df-np 9253
This theorem is referenced by:  genpcl  9280  nqpr  9286  ltexprlem5  9312  reclem2pr  9320  suplem1pr  9324
  Copyright terms: Public domain W3C validator