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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
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