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Theorem recexprlemdisj 7402
Description:  B is disjoint. Lemma for recexpr 7410. (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 7170 . . . . . 6  |-  <Q  Or  Q.
2 ltrelnq 7137 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
31, 2son2lpi 4903 . . . . 5  |-  -.  (
( *Q `  z
)  <Q  ( *Q `  y )  /\  ( *Q `  y )  <Q 
( *Q `  z
) )
4 simprr 504 . . . . . . . . . 10  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  ( *Q `  z )  e.  ( 1st `  A ) )
5 simplr 502 . . . . . . . . . 10  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  ( *Q `  y )  e.  ( 2nd `  A ) )
64, 5jca 302 . . . . . . . . 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 7247 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
8 prltlu 7259 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  z
)  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  ( *Q `  z )  <Q 
( *Q `  y
) )
97, 8syl3an1 1232 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  ( *Q `  z )  e.  ( 1st `  A
)  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( *Q `  z )  <Q  ( *Q `  y ) )
1093expb 1165 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  ( ( *Q `  z )  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) )  ->  ( *Q `  z )  <Q 
( *Q `  y
) )
116, 10sylan2 282 . . . . . . . 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 503 . . . . . . . . . . 11  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  z  <Q  q
)
13 simpll 501 . . . . . . . . . . 11  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  q  <Q  y
)
141, 2sotri 4902 . . . . . . . . . . 11  |-  ( ( z  <Q  q  /\  q  <Q  y )  -> 
z  <Q  y )
1512, 13, 14syl2anc 406 . . . . . . . . . 10  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  z  <Q  y
)
16 ltrnqi 7193 . . . . . . . . . 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 273 . . . . . . . 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 302 . . . . . . 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 114 . . . . . 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 272 . . . . 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 636 . . . 4  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  -.  ( ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  /\  (
z  <Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
2322alrimivv 1829 . . 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 7394 . . . . . . . 8  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
2624recexprlemelu 7395 . . . . . . . 8  |-  ( q  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
2725, 26anbi12i 453 . . . . . . 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 3900 . . . . . . . . . 10  |-  ( y  =  z  ->  (
y  <Q  q  <->  z  <Q  q ) )
29 fveq2 5387 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( *Q `  y )  =  ( *Q `  z
) )
3029eleq1d 2184 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( *Q `  y
)  e.  ( 1st `  A )  <->  ( *Q `  z )  e.  ( 1st `  A ) ) )
3128, 30anbi12d 462 . . . . . . . . 9  |-  ( y  =  z  ->  (
( y  <Q  q  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( z  <Q  q  /\  ( *Q
`  z )  e.  ( 1st `  A
) ) ) )
3231cbvexv 1870 . . . . . . . 8  |-  ( E. y ( y  <Q 
q  /\  ( *Q `  y )  e.  ( 1st `  A ) )  <->  E. z ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) )
3332anbi2i 450 . . . . . . 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 183 . . . . . 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 1882 . . . . . 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 186 . . . . 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 640 . . . 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 1458 . . . . . 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 1429 . . . . 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 1458 . . . . 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 183 . . . 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 186 . . 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 133 . 2  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  B
)  /\  q  e.  ( 2nd `  B ) ) )
4443ralrimiva 2480 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 103   A.wal 1312    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   A.wral 2391   <.cop 3498   class class class wbr 3897   ` cfv 5091   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052   *Qcrq 7056    <Q cltq 7057   P.cnp 7063
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-mi 7078  df-lti 7079  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-inp 7238
This theorem is referenced by:  recexprlempr  7404
  Copyright terms: Public domain W3C validator