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Theorem recexprlemdisj 7571
Description: B is disjoint. Lemma for recexpr 7579. (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 7339 . . . . . 6 |- <Q Or Q.
2 ltrelnq 7306 . . . . . 6 |- <Q C_ ( Q. X. Q. )
31, 2son2lpi 5000 . . . . 5 |- -. ( ( *Q ` z ) <Q ( *Q ` y ) /\ ( *Q ` y ) <Q ( *Q ` z ) )
4 simprr 522 . . . . . . . . . 10 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> ( *Q ` z ) e. ( 1st ` A ) )
5 simplr 520 . . . . . . . . . 10 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> ( *Q ` y ) e. ( 2nd ` A ) )
64, 5jca 304 . . . . . . . . 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 7416 . . . . . . . . . . 11 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
8 prltlu 7428 . . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
97, 8syl3an1 1261 . . . . . . . . . 10 |- ( ( A e. P. /\ ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
1093expb 1194 . . . . . . . . 9 |- ( ( A e. P. /\ ( ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
116, 10sylan2 284 . . . . . . . 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 521 . . . . . . . . . . 11 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> z <Q q )
13 simpll 519 . . . . . . . . . . 11 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> q <Q y )
141, 2sotri 4999 . . . . . . . . . . 11 |- ( ( z <Q q /\ q <Q y ) -> z <Q y )
1512, 13, 14syl2anc 409 . . . . . . . . . 10 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> z <Q y )
16 ltrnqi 7362 . . . . . . . . . 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 275 . . . . . . . 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 304 . . . . . . 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 274 . . . . 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 654 . . . 4 |- ( ( A e. P. /\ q e. Q. ) -> -. ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) )
2322alrimivv 1863 . . 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 7563 . . . . . . . 8 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
2624recexprlemelu 7564 . . . . . . . 8 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
2725, 26anbi12i 456 . . . . . . 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 3985 . . . . . . . . . 10 |- ( y = z -> ( y <Q q <-> z <Q q ) )
29 fveq2 5486 . . . . . . . . . . 11 |- ( y = z -> ( *Q ` y ) = ( *Q ` z ) )
3029eleq1d 2235 . . . . . . . . . 10 |- ( y = z -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` z ) e. ( 1st ` A ) ) )
3128, 30anbi12d 465 . . . . . . . . 9 |- ( y = z -> ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) )
3231cbvexv 1906 . . . . . . . 8 |- ( E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> E. z ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) )
3332anbi2i 453 . . . . . . 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 1920 . . . . . 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 658 . . . 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 1487 . . . . . 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 1458 . . . . 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 1487 . . . . 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 2539 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 1341   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   A.wral 2444   <.cop 3579   class class class wbr 3982   ` cfv 5188   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   *Qcrq 7225   <Q cltq 7226   P.cnp 7232
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-mi 7247  df-lti 7248  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-inp 7407
This theorem is referenced by:  recexprlempr  7573
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