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Theorem recexprlemdisj 7690
Description:  B is disjoint. Lemma for recexpr 7698. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
Assertion
Ref Expression
recexprlemdisj  |-  ( A  e.  P.  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  B
)  /\  q  e.  ( 2nd `  B ) ) )
Distinct variable groups:    x, q, y, A    B, q, x, y

Proof of Theorem recexprlemdisj
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ltsonq 7458 . . . . . 6  |-  <Q  Or  Q.
2 ltrelnq 7425 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
31, 2son2lpi 5062 . . . . 5  |-  -.  (
( *Q `  z
)  <Q  ( *Q `  y )  /\  ( *Q `  y )  <Q 
( *Q `  z
) )
4 simprr 531 . . . . . . . . . 10  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  ( *Q `  z )  e.  ( 1st `  A ) )
5 simplr 528 . . . . . . . . . 10  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  ( *Q `  y )  e.  ( 2nd `  A ) )
64, 5jca 306 . . . . . . . . 9  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  ( ( *Q
`  z )  e.  ( 1st `  A
)  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) )
7 prop 7535 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
8 prltlu 7547 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  z
)  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  ( *Q `  z )  <Q 
( *Q `  y
) )
97, 8syl3an1 1282 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  ( *Q `  z )  e.  ( 1st `  A
)  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( *Q `  z )  <Q  ( *Q `  y ) )
1093expb 1206 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  ( ( *Q `  z )  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) )  ->  ( *Q `  z )  <Q 
( *Q `  y
) )
116, 10sylan2 286 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )  ->  ( *Q `  z )  <Q  ( *Q `  y ) )
12 simprl 529 . . . . . . . . . . 11  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  z  <Q  q
)
13 simpll 527 . . . . . . . . . . 11  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  q  <Q  y
)
141, 2sotri 5061 . . . . . . . . . . 11  |-  ( ( z  <Q  q  /\  q  <Q  y )  -> 
z  <Q  y )
1512, 13, 14syl2anc 411 . . . . . . . . . 10  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  z  <Q  y
)
16 ltrnqi 7481 . . . . . . . . . 10  |-  ( z 
<Q  y  ->  ( *Q
`  y )  <Q 
( *Q `  z
) )
1715, 16syl 14 . . . . . . . . 9  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  ( *Q `  y )  <Q  ( *Q `  z ) )
1817adantl 277 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )  ->  ( *Q `  y )  <Q  ( *Q `  z ) )
1911, 18jca 306 . . . . . . 7  |-  ( ( A  e.  P.  /\  ( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )  ->  ( ( *Q
`  z )  <Q 
( *Q `  y
)  /\  ( *Q `  y )  <Q  ( *Q `  z ) ) )
2019ex 115 . . . . . 6  |-  ( A  e.  P.  ->  (
( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) )  -> 
( ( *Q `  z )  <Q  ( *Q `  y )  /\  ( *Q `  y ) 
<Q  ( *Q `  z
) ) ) )
2120adantr 276 . . . . 5  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  ( ( ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  /\  (
z  <Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) )  -> 
( ( *Q `  z )  <Q  ( *Q `  y )  /\  ( *Q `  y ) 
<Q  ( *Q `  z
) ) ) )
223, 21mtoi 665 . . . 4  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  -.  ( ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  /\  (
z  <Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
2322alrimivv 1886 . . 3  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  A. y A. z  -.  ( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
24 recexpr.1 . . . . . . . . 9  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
2524recexprlemell 7682 . . . . . . . 8  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
2624recexprlemelu 7683 . . . . . . . 8  |-  ( q  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
2725, 26anbi12i 460 . . . . . . 7  |-  ( ( q  e.  ( 1st `  B )  /\  q  e.  ( 2nd `  B
) )  <->  ( E. y ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  E. y
( y  <Q  q  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) ) )
28 breq1 4032 . . . . . . . . . 10  |-  ( y  =  z  ->  (
y  <Q  q  <->  z  <Q  q ) )
29 fveq2 5554 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( *Q `  y )  =  ( *Q `  z
) )
3029eleq1d 2262 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( *Q `  y
)  e.  ( 1st `  A )  <->  ( *Q `  z )  e.  ( 1st `  A ) ) )
3128, 30anbi12d 473 . . . . . . . . 9  |-  ( y  =  z  ->  (
( y  <Q  q  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( z  <Q  q  /\  ( *Q
`  z )  e.  ( 1st `  A
) ) ) )
3231cbvexv 1930 . . . . . . . 8  |-  ( E. y ( y  <Q 
q  /\  ( *Q `  y )  e.  ( 1st `  A ) )  <->  E. z ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) )
3332anbi2i 457 . . . . . . 7  |-  ( ( E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  /\  E. y ( y  <Q 
q  /\  ( *Q `  y )  e.  ( 1st `  A ) ) )  <->  ( E. y ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  E. z
( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) ) )
3427, 33bitri 184 . . . . . 6  |-  ( ( q  e.  ( 1st `  B )  /\  q  e.  ( 2nd `  B
) )  <->  ( E. y ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  E. z
( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) ) )
35 eeanv 1948 . . . . . 6  |-  ( E. y E. z ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  <-> 
( E. y ( q  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  /\  E. z ( z  <Q 
q  /\  ( *Q `  z )  e.  ( 1st `  A ) ) ) )
3634, 35bitr4i 187 . . . . 5  |-  ( ( q  e.  ( 1st `  B )  /\  q  e.  ( 2nd `  B
) )  <->  E. y E. z ( ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  /\  (
z  <Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
3736notbii 669 . . . 4  |-  ( -.  ( q  e.  ( 1st `  B )  /\  q  e.  ( 2nd `  B ) )  <->  -.  E. y E. z ( ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  /\  (
z  <Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
38 alnex 1510 . . . . . 6  |-  ( A. z  -.  ( ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  /\  (
z  <Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) )  <->  -.  E. z
( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
3938albii 1481 . . . . 5  |-  ( A. y A. z  -.  (
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  <->  A. y  -.  E. z
( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
40 alnex 1510 . . . . 5  |-  ( A. y  -.  E. z ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  <->  -.  E. y E. z
( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
4139, 40bitri 184 . . . 4  |-  ( A. y A. z  -.  (
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  <->  -.  E. y E. z
( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
4237, 41bitr4i 187 . . 3  |-  ( -.  ( q  e.  ( 1st `  B )  /\  q  e.  ( 2nd `  B ) )  <->  A. y A. z  -.  ( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  /\  ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
4323, 42sylibr 134 . 2  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  B
)  /\  q  e.  ( 2nd `  B ) ) )
4443ralrimiva 2567 1  |-  ( A  e.  P.  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  B
)  /\  q  e.  ( 2nd `  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104   A.wal 1362    = wceq 1364   E.wex 1503    e. wcel 2164   {cab 2179   A.wral 2472   <.cop 3621   class class class wbr 4029   ` cfv 5254   1stc1st 6191   2ndc2nd 6192   Q.cnq 7340   *Qcrq 7344    <Q cltq 7345   P.cnp 7351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-mi 7366  df-lti 7367  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-inp 7526
This theorem is referenced by:  recexprlempr  7692
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