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Theorem recexprlemdisj 7628
Description:  B is disjoint. Lemma for recexpr 7636. (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 7396 . . . . . 6  |-  <Q  Or  Q.
2 ltrelnq 7363 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
31, 2son2lpi 5025 . . . . 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 7473 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
8 prltlu 7485 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  z
)  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  ( *Q `  z )  <Q 
( *Q `  y
) )
97, 8syl3an1 1271 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  ( *Q `  z )  e.  ( 1st `  A
)  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( *Q `  z )  <Q  ( *Q `  y ) )
1093expb 1204 . . . . . . . . 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 5024 . . . . . . . . . . 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 7419 . . . . . . . . . 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 664 . . . 4  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  -.  ( ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  /\  (
z  <Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
2322alrimivv 1875 . . 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 7620 . . . . . . . 8  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
2624recexprlemelu 7621 . . . . . . . 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 4006 . . . . . . . . . 10  |-  ( y  =  z  ->  (
y  <Q  q  <->  z  <Q  q ) )
29 fveq2 5515 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( *Q `  y )  =  ( *Q `  z
) )
3029eleq1d 2246 . . . . . . . . . 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 1918 . . . . . . . 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 1932 . . . . . 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 668 . . . 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 1499 . . . . . 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 1470 . . . . 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 1499 . . . . 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 2550 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 1351    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   A.wral 2455   <.cop 3595   class class class wbr 4003   ` cfv 5216   1stc1st 6138   2ndc2nd 6139   Q.cnq 7278   *Qcrq 7282    <Q cltq 7283   P.cnp 7289
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-mi 7304  df-lti 7305  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-inp 7464
This theorem is referenced by:  recexprlempr  7630
  Copyright terms: Public domain W3C validator