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Theorem elnp1st2nd 6798
Description: Membership in positive reals, using  1st and  2nd to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.)
Assertion
Ref Expression
elnp1st2nd  |-  ( A  e.  P.  <->  ( ( A  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e. 
Q.  q  e.  ( 1st `  A )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  A ) ) )  /\  (
( A. q  e. 
Q.  ( q  e.  ( 1st `  A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  A
)  \/  r  e.  ( 2nd `  A
) ) ) ) ) )
Distinct variable group:    r, q, A

Proof of Theorem elnp1st2nd
StepHypRef Expression
1 npsspw 6793 . . . . 5  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
21sseli 3004 . . . 4  |-  ( A  e.  P.  ->  A  e.  ( ~P Q.  X.  ~P Q. ) )
3 prop 6797 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
4 elinp 6796 . . . . . . 7  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  <->  ( ( ( ( 1st `  A )  C_  Q.  /\  ( 2nd `  A
)  C_  Q. )  /\  ( E. q  e. 
Q.  q  e.  ( 1st `  A )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  A ) ) )  /\  (
( A. q  e. 
Q.  ( q  e.  ( 1st `  A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  A
)  \/  r  e.  ( 2nd `  A
) ) ) ) ) )
53, 4sylib 120 . . . . . 6  |-  ( A  e.  P.  ->  (
( ( ( 1st `  A )  C_  Q.  /\  ( 2nd `  A
)  C_  Q. )  /\  ( E. q  e. 
Q.  q  e.  ( 1st `  A )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  A ) ) )  /\  (
( A. q  e. 
Q.  ( q  e.  ( 1st `  A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  A
)  \/  r  e.  ( 2nd `  A
) ) ) ) ) )
65simpld 110 . . . . 5  |-  ( A  e.  P.  ->  (
( ( 1st `  A
)  C_  Q.  /\  ( 2nd `  A )  C_  Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A
) ) ) )
76simprd 112 . . . 4  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A
) ) )
82, 7jca 300 . . 3  |-  ( A  e.  P.  ->  ( A  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e. 
Q.  q  e.  ( 1st `  A )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  A ) ) ) )
95simprd 112 . . 3  |-  ( A  e.  P.  ->  (
( A. q  e. 
Q.  ( q  e.  ( 1st `  A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  A
)  \/  r  e.  ( 2nd `  A
) ) ) ) )
108, 9jca 300 . 2  |-  ( A  e.  P.  ->  (
( A  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A
) ) )  /\  ( ( A. q  e.  Q.  ( q  e.  ( 1st `  A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  A
)  \/  r  e.  ( 2nd `  A
) ) ) ) ) )
11 1st2nd2 5853 . . . 4  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
1211ad2antrr 472 . . 3  |-  ( ( ( A  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A
) ) )  /\  ( ( A. q  e.  Q.  ( q  e.  ( 1st `  A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  A
)  \/  r  e.  ( 2nd `  A
) ) ) ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
13 xp1st 5844 . . . . . . . 8  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 1st `  A
)  e.  ~P Q. )
1413elpwid 3410 . . . . . . 7  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 1st `  A
)  C_  Q. )
15 xp2nd 5845 . . . . . . . 8  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 2nd `  A
)  e.  ~P Q. )
1615elpwid 3410 . . . . . . 7  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 2nd `  A
)  C_  Q. )
1714, 16jca 300 . . . . . 6  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( ( 1st `  A
)  C_  Q.  /\  ( 2nd `  A )  C_  Q. ) )
1817anim1i 333 . . . . 5  |-  ( ( A  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  A ) ) )  ->  (
( ( 1st `  A
)  C_  Q.  /\  ( 2nd `  A )  C_  Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A
) ) ) )
1918anim1i 333 . . . 4  |-  ( ( ( A  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A
) ) )  /\  ( ( A. q  e.  Q.  ( q  e.  ( 1st `  A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  A
)  \/  r  e.  ( 2nd `  A
) ) ) ) )  ->  ( (
( ( 1st `  A
)  C_  Q.  /\  ( 2nd `  A )  C_  Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A
) ) )  /\  ( ( A. q  e.  Q.  ( q  e.  ( 1st `  A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  A
)  \/  r  e.  ( 2nd `  A
) ) ) ) ) )
2019, 4sylibr 132 . . 3  |-  ( ( ( A  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A
) ) )  /\  ( ( A. q  e.  Q.  ( q  e.  ( 1st `  A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  A
)  \/  r  e.  ( 2nd `  A
) ) ) ) )  ->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  e.  P. )
2112, 20eqeltrd 2159 . 2  |-  ( ( ( A  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A
) ) )  /\  ( ( A. q  e.  Q.  ( q  e.  ( 1st `  A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  A
)  \/  r  e.  ( 2nd `  A
) ) ) ) )  ->  A  e.  P. )
2210, 21impbii 124 1  |-  ( A  e.  P.  <->  ( ( A  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e. 
Q.  q  e.  ( 1st `  A )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  A ) ) )  /\  (
( A. q  e. 
Q.  ( q  e.  ( 1st `  A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  A
)  \/  r  e.  ( 2nd `  A
) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2353   E.wrex 2354    C_ wss 2982   ~Pcpw 3400   <.cop 3419   class class class wbr 3805    X. cxp 4389   ` cfv 4952   1stc1st 5817   2ndc2nd 5818   Q.cnq 6602    <Q cltq 6607   P.cnp 6613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-1st 5819  df-2nd 5820  df-qs 6200  df-ni 6626  df-nqqs 6670  df-inp 6788
This theorem is referenced by:  addclpr  6859  mulclpr  6894  ltexprlempr  6930  recexprlempr  6954  cauappcvgprlemcl  6975  caucvgprlemcl  6998  caucvgprprlemcl  7026
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