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Theorem elnp1st2nd 7396
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 7391 . . . . 5  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
21sseli 3124 . . . 4  |-  ( A  e.  P.  ->  A  e.  ( ~P Q.  X.  ~P Q. ) )
3 prop 7395 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
4 elinp 7394 . . . . . . 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 121 . . . . . 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 111 . . . . 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 113 . . . 4  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A
) ) )
82, 7jca 304 . . 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 113 . . 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 304 . 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 6123 . . . 4  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
1211ad2antrr 480 . . 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 6113 . . . . . . . 8  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 1st `  A
)  e.  ~P Q. )
1413elpwid 3554 . . . . . . 7  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 1st `  A
)  C_  Q. )
15 xp2nd 6114 . . . . . . . 8  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 2nd `  A
)  e.  ~P Q. )
1615elpwid 3554 . . . . . . 7  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 2nd `  A
)  C_  Q. )
1714, 16jca 304 . . . . . 6  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( ( 1st `  A
)  C_  Q.  /\  ( 2nd `  A )  C_  Q. ) )
1817anim1i 338 . . . . 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 338 . . . 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 133 . . 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 2234 . 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 125 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 103    <-> wb 104    \/ wo 698    /\ w3a 963    = wceq 1335    e. wcel 2128   A.wral 2435   E.wrex 2436    C_ wss 3102   ~Pcpw 3543   <.cop 3563   class class class wbr 3965    X. cxp 4584   ` cfv 5170   1stc1st 6086   2ndc2nd 6087   Q.cnq 7200    <Q cltq 7205   P.cnp 7211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-1st 6088  df-2nd 6089  df-qs 6486  df-ni 7224  df-nqqs 7268  df-inp 7386
This theorem is referenced by:  addclpr  7457  mulclpr  7492  ltexprlempr  7528  recexprlempr  7552  cauappcvgprlemcl  7573  caucvgprlemcl  7596  caucvgprprlemcl  7624
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