ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recexprlemdisj Unicode version

Theorem recexprlemdisj 6786
Description:  B is disjoint. Lemma for recexpr 6794. (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 6554 . . . . . 6  |-  <Q  Or  Q.
2 ltrelnq 6521 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
31, 2son2lpi 4749 . . . . 5  |-  -.  (
( *Q `  z
)  <Q  ( *Q `  y )  /\  ( *Q `  y )  <Q 
( *Q `  z
) )
4 simprr 492 . . . . . . . . . 10  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  ( *Q `  z )  e.  ( 1st `  A ) )
5 simplr 490 . . . . . . . . . 10  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  ( *Q `  y )  e.  ( 2nd `  A ) )
64, 5jca 294 . . . . . . . . 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 6631 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
8 prltlu 6643 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  z
)  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  ( *Q `  z )  <Q 
( *Q `  y
) )
97, 8syl3an1 1179 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  ( *Q `  z )  e.  ( 1st `  A
)  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( *Q `  z )  <Q  ( *Q `  y ) )
1093expb 1116 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  ( ( *Q `  z )  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) )  ->  ( *Q `  z )  <Q 
( *Q `  y
) )
116, 10sylan2 274 . . . . . . . 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 491 . . . . . . . . . . 11  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  z  <Q  q
)
13 simpll 489 . . . . . . . . . . 11  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  q  <Q  y
)
141, 2sotri 4748 . . . . . . . . . . 11  |-  ( ( z  <Q  q  /\  q  <Q  y )  -> 
z  <Q  y )
1512, 13, 14syl2anc 397 . . . . . . . . . 10  |-  ( ( ( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  /\  ( z  <Q  q  /\  ( *Q `  z
)  e.  ( 1st `  A ) ) )  ->  z  <Q  y
)
16 ltrnqi 6577 . . . . . . . . . 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 266 . . . . . . . 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 294 . . . . . . 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 112 . . . . . 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 265 . . . . 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 600 . . . 4  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  -.  ( ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  /\  (
z  <Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) ) )
2322alrimivv 1771 . . 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 6778 . . . . . . . 8  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
2624recexprlemelu 6779 . . . . . . . 8  |-  ( q  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
2725, 26anbi12i 441 . . . . . . 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 3795 . . . . . . . . . 10  |-  ( y  =  z  ->  (
y  <Q  q  <->  z  <Q  q ) )
29 fveq2 5206 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( *Q `  y )  =  ( *Q `  z
) )
3029eleq1d 2122 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( *Q `  y
)  e.  ( 1st `  A )  <->  ( *Q `  z )  e.  ( 1st `  A ) ) )
3128, 30anbi12d 450 . . . . . . . . 9  |-  ( y  =  z  ->  (
( y  <Q  q  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( z  <Q  q  /\  ( *Q
`  z )  e.  ( 1st `  A
) ) ) )
3231cbvexv 1811 . . . . . . . 8  |-  ( E. y ( y  <Q 
q  /\  ( *Q `  y )  e.  ( 1st `  A ) )  <->  E. z ( z 
<Q  q  /\  ( *Q `  z )  e.  ( 1st `  A
) ) )
3332anbi2i 438 . . . . . . 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 177 . . . . . 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 1823 . . . . . 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 180 . . . . 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 604 . . . 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 1404 . . . . . 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 1375 . . . . 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 1404 . . . . 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 177 . . . 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 180 . . 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 141 . 2  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  B
)  /\  q  e.  ( 2nd `  B ) ) )
4443ralrimiva 2409 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 101   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042   A.wral 2323   <.cop 3406   class class class wbr 3792   ` cfv 4930   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436   *Qcrq 6440    <Q cltq 6441   P.cnp 6447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-mi 6462  df-lti 6463  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-inp 6622
This theorem is referenced by:  recexprlempr  6788
  Copyright terms: Public domain W3C validator