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Theorem elnp1st2nd 6602
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 6597 . . . . 5  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
21sseli 2966 . . . 4  |-  ( A  e.  P.  ->  A  e.  ( ~P Q.  X.  ~P Q. ) )
3 prop 6601 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
4 elinp 6600 . . . . . . 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 131 . . . . . 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 109 . . . . 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 111 . . . 4  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A
) ) )
82, 7jca 294 . . 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 111 . . 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 294 . 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 5826 . . . 4  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
1211ad2antrr 465 . . 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 5817 . . . . . . . 8  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 1st `  A
)  e.  ~P Q. )
1413elpwid 3394 . . . . . . 7  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 1st `  A
)  C_  Q. )
15 xp2nd 5818 . . . . . . . 8  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 2nd `  A
)  e.  ~P Q. )
1615elpwid 3394 . . . . . . 7  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 2nd `  A
)  C_  Q. )
1714, 16jca 294 . . . . . 6  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( ( 1st `  A
)  C_  Q.  /\  ( 2nd `  A )  C_  Q. ) )
1817anim1i 327 . . . . 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 327 . . . 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 141 . . 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 2128 . 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 121 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 101    <-> wb 102    \/ wo 637    /\ w3a 894    = wceq 1257    e. wcel 1407   A.wral 2321   E.wrex 2322    C_ wss 2942   ~Pcpw 3384   <.cop 3403   class class class wbr 3789    X. cxp 4368   ` cfv 4927   1stc1st 5790   2ndc2nd 5791   Q.cnq 6406    <Q cltq 6411   P.cnp 6417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-id 4055  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-1st 5792  df-2nd 5793  df-qs 6140  df-ni 6430  df-nqqs 6474  df-inp 6592
This theorem is referenced by:  addclpr  6663  mulclpr  6698  ltexprlempr  6734  recexprlempr  6758  cauappcvgprlemcl  6779  caucvgprlemcl  6802  caucvgprprlemcl  6830
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