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Theorem elnp1st2nd 7438
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 7433 . . . . 5  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
21sseli 3143 . . . 4  |-  ( A  e.  P.  ->  A  e.  ( ~P Q.  X.  ~P Q. ) )
3 prop 7437 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
4 elinp 7436 . . . . . . 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 6154 . . . 4  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
1211ad2antrr 485 . . 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 6144 . . . . . . . 8  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 1st `  A
)  e.  ~P Q. )
1413elpwid 3577 . . . . . . 7  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 1st `  A
)  C_  Q. )
15 xp2nd 6145 . . . . . . . 8  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 2nd `  A
)  e.  ~P Q. )
1615elpwid 3577 . . . . . . 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 2247 . 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 703    /\ w3a 973    = wceq 1348    e. wcel 2141   A.wral 2448   E.wrex 2449    C_ wss 3121   ~Pcpw 3566   <.cop 3586   class class class wbr 3989    X. cxp 4609   ` cfv 5198   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242    <Q cltq 7247   P.cnp 7253
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-qs 6519  df-ni 7266  df-nqqs 7310  df-inp 7428
This theorem is referenced by:  addclpr  7499  mulclpr  7534  ltexprlempr  7570  recexprlempr  7594  cauappcvgprlemcl  7615  caucvgprlemcl  7638  caucvgprprlemcl  7666
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