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Theorem recexprlemdisj 6728
Description: B is disjoint. Lemma for recexpr 6736. (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 6496 . . . . . 6 |- <Q Or Q.
2 ltrelnq 6463 . . . . . 6 |- <Q C_ ( Q. X. Q. )
31, 2son2lpi 4721 . . . . 5 |- -. ( ( *Q ` z ) <Q ( *Q ` y ) /\ ( *Q ` y ) <Q ( *Q ` z ) )
4 simprr 484 . . . . . . . . . 10 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> ( *Q ` z ) e. ( 1st ` A ) )
5 simplr 482 . . . . . . . . . 10 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> ( *Q ` y ) e. ( 2nd ` A ) )
64, 5jca 290 . . . . . . . . 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 6573 . . . . . . . . . . 11 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
8 prltlu 6585 . . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
97, 8syl3an1 1168 . . . . . . . . . 10 |- ( ( A e. P. /\ ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
1093expb 1105 . . . . . . . . 9 |- ( ( A e. P. /\ ( ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
116, 10sylan2 270 . . . . . . . 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 483 . . . . . . . . . . 11 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> z <Q q )
13 simpll 481 . . . . . . . . . . 11 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> q <Q y )
141, 2sotri 4720 . . . . . . . . . . 11 |- ( ( z <Q q /\ q <Q y ) -> z <Q y )
1512, 13, 14syl2anc 391 . . . . . . . . . 10 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> z <Q y )
16 ltrnqi 6519 . . . . . . . . . 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 262 . . . . . . . 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 290 . . . . . . 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 108 . . . . . 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 261 . . . . 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 590 . . . 4 |- ( ( A e. P. /\ q e. Q. ) -> -. ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) )
2322alrimivv 1755 . . 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 6720 . . . . . . . 8 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
2624recexprlemelu 6721 . . . . . . . 8 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
2725, 26anbi12i 433 . . . . . . 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 3767 . . . . . . . . . 10 |- ( y = z -> ( y <Q q <-> z <Q q ) )
29 fveq2 5178 . . . . . . . . . . 11 |- ( y = z -> ( *Q ` y ) = ( *Q ` z ) )
3029eleq1d 2106 . . . . . . . . . 10 |- ( y = z -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` z ) e. ( 1st ` A ) ) )
3128, 30anbi12d 442 . . . . . . . . 9 |- ( y = z -> ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) )
3231cbvexv 1795 . . . . . . . 8 |- ( E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> E. z ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) )
3332anbi2i 430 . . . . . . 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 173 . . . . . 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 1807 . . . . . 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 176 . . . . 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 594 . . . 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 1388 . . . . . 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 1359 . . . . 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 1388 . . . . 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 173 . . . 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 176 . . 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 137 . 2 |- ( ( A e. P. /\ q e. Q. ) -> -. ( q e. ( 1st ` B ) /\ q e. ( 2nd ` B ) ) )
4443ralrimiva 2392 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 97   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026   A.wral 2306   <.cop 3378   class class class wbr 3764   ` cfv 4902   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378   *Qcrq 6382   <Q cltq 6383   P.cnp 6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-lti 6405  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564
This theorem is referenced by:  recexprlempr  6730
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