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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 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
Assertion
Ref Expression
recexprlemdisj (𝐴P → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)))
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 ⊆ (Q × Q)
31, 2son2lpi 4721 . . . . 5 ¬ ((*Q𝑧) <Q (*Q𝑦) ∧ (*Q𝑦) <Q (*Q𝑧))
4 simprr 484 . . . . . . . . . 10 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → (*Q𝑧) ∈ (1st𝐴))
5 simplr 482 . . . . . . . . . 10 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → (*Q𝑦) ∈ (2nd𝐴))
64, 5jca 290 . . . . . . . . 9 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → ((*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)))
7 prop 6573 . . . . . . . . . . 11 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
8 prltlu 6585 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → (*Q𝑧) <Q (*Q𝑦))
97, 8syl3an1 1168 . . . . . . . . . 10 ((𝐴P ∧ (*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → (*Q𝑧) <Q (*Q𝑦))
1093expb 1105 . . . . . . . . 9 ((𝐴P ∧ ((*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴))) → (*Q𝑧) <Q (*Q𝑦))
116, 10sylan2 270 . . . . . . . 8 ((𝐴P ∧ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴)))) → (*Q𝑧) <Q (*Q𝑦))
12 simprl 483 . . . . . . . . . . 11 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → 𝑧 <Q 𝑞)
13 simpll 481 . . . . . . . . . . 11 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → 𝑞 <Q 𝑦)
141, 2sotri 4720 . . . . . . . . . . 11 ((𝑧 <Q 𝑞𝑞 <Q 𝑦) → 𝑧 <Q 𝑦)
1512, 13, 14syl2anc 391 . . . . . . . . . 10 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
16 ltrnqi 6519 . . . . . . . . . 10 (𝑧 <Q 𝑦 → (*Q𝑦) <Q (*Q𝑧))
1715, 16syl 14 . . . . . . . . 9 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → (*Q𝑦) <Q (*Q𝑧))
1817adantl 262 . . . . . . . 8 ((𝐴P ∧ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴)))) → (*Q𝑦) <Q (*Q𝑧))
1911, 18jca 290 . . . . . . 7 ((𝐴P ∧ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴)))) → ((*Q𝑧) <Q (*Q𝑦) ∧ (*Q𝑦) <Q (*Q𝑧)))
2019ex 108 . . . . . 6 (𝐴P → (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → ((*Q𝑧) <Q (*Q𝑦) ∧ (*Q𝑦) <Q (*Q𝑧))))
2120adantr 261 . . . . 5 ((𝐴P𝑞Q) → (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → ((*Q𝑧) <Q (*Q𝑦) ∧ (*Q𝑦) <Q (*Q𝑧))))
223, 21mtoi 590 . . . 4 ((𝐴P𝑞Q) → ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
2322alrimivv 1755 . . 3 ((𝐴P𝑞Q) → ∀𝑦𝑧 ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
24 recexpr.1 . . . . . . . . 9 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
2524recexprlemell 6720 . . . . . . . 8 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
2624recexprlemelu 6721 . . . . . . . 8 (𝑞 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)))
2725, 26anbi12i 433 . . . . . . 7 ((𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴))))
28 breq1 3767 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑦 <Q 𝑞𝑧 <Q 𝑞))
29 fveq2 5178 . . . . . . . . . . 11 (𝑦 = 𝑧 → (*Q𝑦) = (*Q𝑧))
3029eleq1d 2106 . . . . . . . . . 10 (𝑦 = 𝑧 → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q𝑧) ∈ (1st𝐴)))
3128, 30anbi12d 442 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3231cbvexv 1795 . . . . . . . 8 (∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ ∃𝑧(𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴)))
3332anbi2i 430 . . . . . . 7 ((∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴))) ↔ (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ ∃𝑧(𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3427, 33bitri 173 . . . . . 6 ((𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ ∃𝑧(𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
35 eeanv 1807 . . . . . 6 (∃𝑦𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) ↔ (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ ∃𝑧(𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3634, 35bitr4i 176 . . . . 5 ((𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ ∃𝑦𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3736notbii 594 . . . 4 (¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ ¬ ∃𝑦𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
38 alnex 1388 . . . . . 6 (∀𝑧 ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) ↔ ¬ ∃𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3938albii 1359 . . . . 5 (∀𝑦𝑧 ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) ↔ ∀𝑦 ¬ ∃𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
40 alnex 1388 . . . . 5 (∀𝑦 ¬ ∃𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) ↔ ¬ ∃𝑦𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
4139, 40bitri 173 . . . 4 (∀𝑦𝑧 ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) ↔ ¬ ∃𝑦𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
4237, 41bitr4i 176 . . 3 (¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ ∀𝑦𝑧 ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
4323, 42sylibr 137 . 2 ((𝐴P𝑞Q) → ¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)))
4443ralrimiva 2392 1 (𝐴P → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wal 1241   = wceq 1243  wex 1381  wcel 1393  {cab 2026  wral 2306  cop 3378   class class class wbr 3764  cfv 4902  1st c1st 5765  2nd c2nd 5766  Qcnq 6378  *Qcrq 6382   <Q cltq 6383  Pcnp 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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