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Theorem recexprlemdisj 6820
Description: B is disjoint. Lemma for recexpr 6828. (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 6588 . . . . . 6 |- <Q Or Q.
2 ltrelnq 6555 . . . . . 6 |- <Q C_ ( Q. X. Q. )
31, 2son2lpi 4741 . . . . 5 |- -. ( ( *Q ` z ) <Q ( *Q ` y ) /\ ( *Q ` y ) <Q ( *Q ` z ) )
4 simprr 498 . . . . . . . . . 10 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> ( *Q ` z ) e. ( 1st ` A ) )
5 simplr 496 . . . . . . . . . 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 6665 . . . . . . . . . . 11 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
8 prltlu 6677 . . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
97, 8syl3an1 1202 . . . . . . . . . 10 |- ( ( A e. P. /\ ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
1093expb 1139 . . . . . . . . 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 497 . . . . . . . . . . 11 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> z <Q q )
13 simpll 495 . . . . . . . . . . 11 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> q <Q y )
141, 2sotri 4740 . . . . . . . . . . 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 6611 . . . . . . . . . 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 622 . . . 4 |- ( ( A e. P. /\ q e. Q. ) -> -. ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) )
2322alrimivv 1796 . . 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 6812 . . . . . . . 8 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
2624recexprlemelu 6813 . . . . . . . 8 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
2725, 26anbi12i 447 . . . . . . 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 3788 . . . . . . . . . 10 |- ( y = z -> ( y <Q q <-> z <Q q ) )
29 fveq2 5198 . . . . . . . . . . 11 |- ( y = z -> ( *Q ` y ) = ( *Q ` z ) )
3029eleq1d 2147 . . . . . . . . . 10 |- ( y = z -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` z ) e. ( 1st ` A ) ) )
3128, 30anbi12d 456 . . . . . . . . 9 |- ( y = z -> ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) )
3231cbvexv 1836 . . . . . . . 8 |- ( E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> E. z ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) )
3332anbi2i 444 . . . . . . 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 1848 . . . . . 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 626 . . . 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 1428 . . . . . 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 1399 . . . . 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 1428 . . . . 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 2434 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 1282   = wceq 1284   E.wex 1421   e. wcel 1433   {cab 2067   A.wral 2348   <.cop 3401   class class class wbr 3785   ` cfv 4922   1stc1st 5785   2ndc2nd 5786   Q.cnq 6470   *Qcrq 6474   <Q cltq 6475   P.cnp 6481
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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329
This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-mi 6496  df-lti 6497  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-inp 6656
This theorem is referenced by:  recexprlempr  6822
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