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Theorem recexprlemdisj 7592
Description: B is disjoint. Lemma for recexpr 7600. (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 7360 . . . . . 6 |- <Q Or Q.
2 ltrelnq 7327 . . . . . 6 |- <Q C_ ( Q. X. Q. )
31, 2son2lpi 5007 . . . . 5 |- -. ( ( *Q ` z ) <Q ( *Q ` y ) /\ ( *Q ` y ) <Q ( *Q ` z ) )
4 simprr 527 . . . . . . . . . 10 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> ( *Q ` z ) e. ( 1st ` A ) )
5 simplr 525 . . . . . . . . . 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 7437 . . . . . . . . . . 11 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
8 prltlu 7449 . . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
97, 8syl3an1 1266 . . . . . . . . . 10 |- ( ( A e. P. /\ ( *Q ` z ) e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( *Q ` z ) <Q ( *Q ` y ) )
1093expb 1199 . . . . . . . . 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 526 . . . . . . . . . . 11 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> z <Q q )
13 simpll 524 . . . . . . . . . . 11 |- ( ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) -> q <Q y )
141, 2sotri 5006 . . . . . . . . . . 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 7383 . . . . . . . . . 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 659 . . . 4 |- ( ( A e. P. /\ q e. Q. ) -> -. ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) /\ ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) )
2322alrimivv 1868 . . 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 7584 . . . . . . . 8 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
2624recexprlemelu 7585 . . . . . . . 8 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
2725, 26anbi12i 457 . . . . . . 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 3992 . . . . . . . . . 10 |- ( y = z -> ( y <Q q <-> z <Q q ) )
29 fveq2 5496 . . . . . . . . . . 11 |- ( y = z -> ( *Q ` y ) = ( *Q ` z ) )
3029eleq1d 2239 . . . . . . . . . 10 |- ( y = z -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` z ) e. ( 1st ` A ) ) )
3128, 30anbi12d 470 . . . . . . . . 9 |- ( y = z -> ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) ) )
3231cbvexv 1911 . . . . . . . 8 |- ( E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> E. z ( z <Q q /\ ( *Q ` z ) e. ( 1st ` A ) ) )
3332anbi2i 454 . . . . . . 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 1925 . . . . . 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 663 . . . 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 1492 . . . . . 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 1463 . . . . 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 1492 . . . . 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 2543 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 1346   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   A.wral 2448   <.cop 3586   class class class wbr 3989   ` cfv 5198   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   *Qcrq 7246   <Q cltq 7247   P.cnp 7253
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-mi 7268  df-lti 7269  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428
This theorem is referenced by:  recexprlempr  7594
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