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Theorem recexprlemdisj 7168
Description:  B is disjoint. Lemma for recexpr 7176. (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 6936 . . . . . 6  |-  <Q  Or  Q.
2 ltrelnq 6903 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
31, 2son2lpi 4815 . . . . 5  |-  -.  (
( *Q `  z
)  <Q  ( *Q `  y )  /\  ( *Q `  y )  <Q 
( *Q `  z
) )
4 simprr 499 . . . . . . . . . 10  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  ( *Q `  z )  e.  ( 1st `  A ) )
5 simplr 497 . . . . . . . . . 10  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  ( *Q `  y )  e.  ( 2nd `  A ) )
64, 5jca 300 . . . . . . . . 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 7013 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
8 prltlu 7025 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  z
)  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  ( *Q `  z )  <Q 
( *Q `  y
) )
97, 8syl3an1 1207 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  ( *Q `  z )  e.  ( 1st `  A
)  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( *Q `  z )  <Q  ( *Q `  y ) )
1093expb 1144 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  ( ( *Q `  z )  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) )  ->  ( *Q `  z )  <Q 
( *Q `  y
) )
116, 10sylan2 280 . . . . . . . 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 498 . . . . . . . . . . 11  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  z  <Q  q
)
13 simpll 496 . . . . . . . . . . 11  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  q  <Q  y
)
141, 2sotri 4814 . . . . . . . . . . 11  |-  ( ( z  <Q  q  /\  q  <Q  y )  -> 
z  <Q  y )
1512, 13, 14syl2anc 403 . . . . . . . . . 10  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  z  <Q  y
)
16 ltrnqi 6959 . . . . . . . . . 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 271 . . . . . . . 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 300 . . . . . . 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 113 . . . . . 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 270 . . . . 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 625 . . . 4  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  -.  ( ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  /\  (
z  <Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
2322alrimivv 1803 . . 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 7160 . . . . . . . 8  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
2624recexprlemelu 7161 . . . . . . . 8  |-  ( q  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
2725, 26anbi12i 448 . . . . . . 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 3840 . . . . . . . . . 10  |-  ( y  =  z  ->  (
y  <Q  q  <->  z  <Q  q ) )
29 fveq2 5289 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( *Q `  y )  =  ( *Q `  z
) )
3029eleq1d 2156 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( *Q `  y
)  e.  ( 1st `  A )  <->  ( *Q `  z )  e.  ( 1st `  A ) ) )
3128, 30anbi12d 457 . . . . . . . . 9  |-  ( y  =  z  ->  (
( y  <Q  q  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( z  <Q  q  /\  ( *Q
`  z )  e.  ( 1st `  A
) ) ) )
3231cbvexv 1843 . . . . . . . 8  |-  ( E. y ( y  <Q 
q  /\  ( *Q `  y )  e.  ( 1st `  A ) )  <->  E. z ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) )
3332anbi2i 445 . . . . . . 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 182 . . . . . 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 1855 . . . . . 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 185 . . . . 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 629 . . . 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 1433 . . . . . 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 1404 . . . . 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 1433 . . . . 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 182 . . . 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 185 . . 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 132 . 2  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  B
)  /\  q  e.  ( 2nd `  B ) ) )
4443ralrimiva 2446 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 102   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   A.wral 2359   <.cop 3444   class class class wbr 3837   ` cfv 5002   1stc1st 5891   2ndc2nd 5892   Q.cnq 6818   *Qcrq 6822    <Q cltq 6823   P.cnp 6829
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-mi 6844  df-lti 6845  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-inp 7004
This theorem is referenced by:  recexprlempr  7170
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