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Theorem recexprlemdisj 7840
Description: B is disjoint. Lemma for recexpr 7848. (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 7608 . . . . . 6 |- <Q Or Q.
2 ltrelnq 7575 . . . . . 6 |- <Q C_ ( Q. X. Q. )
31, 2son2lpi 5131 . . . . 5 |- -. ( ( *Q ` z ) <Q ( *Q ` y ) /\ ( *Q ` y ) <Q ( *Q ` z ) )
4 simprr 531 . . . . . . . . . 10 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> ( *Q ` z ) e. ( 1st ` A ) )
5 simplr 528 . . . . . . . . . 10 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> ( *Q ` y ) e. ( 2nd ` A ) )
64, 5jca 306 . . . . . . . . 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 7685 . . . . . . . . . . 11 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
8 prltlu 7697 . . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
97, 8syl3an1 1304 . . . . . . . . . 10 |- ( ( A e. P. /\ ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
1093expb 1228 . . . . . . . . 9 |- ( ( A e. P. /\ ( ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
116, 10sylan2 286 . . . . . . . 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 529 . . . . . . . . . . 11 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> z <Q q )
13 simpll 527 . . . . . . . . . . 11 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> q <Q y )
141, 2sotri 5130 . . . . . . . . . . 11 |- ( ( z <Q q /\ q <Q y ) -> z <Q y )
1512, 13, 14syl2anc 411 . . . . . . . . . 10 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> z <Q y )
16 ltrnqi 7631 . . . . . . . . . 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 277 . . . . . . . 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 306 . . . . . . 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 115 . . . . . 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 276 . . . . 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 668 . . . 4 |- ( ( A e. P. /\ q e. Q. ) -> -. ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) )
2322alrimivv 1921 . . 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 7832 . . . . . . . 8 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
2624recexprlemelu 7833 . . . . . . . 8 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
2725, 26anbi12i 460 . . . . . . 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 4089 . . . . . . . . . 10 |- ( y = z -> ( y <Q q <-> z <Q q ) )
29 fveq2 5635 . . . . . . . . . . 11 |- ( y = z -> ( *Q ` y ) = ( *Q ` z ) )
3029eleq1d 2298 . . . . . . . . . 10 |- ( y = z -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` z ) e. ( 1st ` A ) ) )
3128, 30anbi12d 473 . . . . . . . . 9 |- ( y = z -> ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) )
3231cbvexv 1965 . . . . . . . 8 |- ( E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> E. z ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) )
3332anbi2i 457 . . . . . . 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 184 . . . . . 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 1983 . . . . . 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 187 . . . . 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 672 . . . 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 1545 . . . . . 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 1516 . . . . 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 1545 . . . . 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 184 . . . 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 187 . . 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 134 . 2 |- ( ( A e. P. /\ q e. Q. ) -> -. ( q e. ( 1st ` B ) /\ q e. ( 2nd ` B ) ) )
4443ralrimiva 2603 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 104   A.wal 1393   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   A.wral 2508   <.cop 3670   class class class wbr 4086   ` cfv 5324   1stc1st 6296   2ndc2nd 6297   Q.cnq 7490   *Qcrq 7494   <Q cltq 7495   P.cnp 7501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-mi 7516  df-lti 7517  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-inp 7676
This theorem is referenced by:  recexprlempr  7842
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