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Theorem recexprlemdisj 6882
Description: 𝐵 is disjoint. Lemma for recexpr 6890. (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 6650 . . . . . 6 <Q Or Q
2 ltrelnq 6617 . . . . . 6 <Q ⊆ (Q × Q)
31, 2son2lpi 4751 . . . . 5 ¬ ((*Q𝑧) <Q (*Q𝑦) ∧ (*Q𝑦) <Q (*Q𝑧))
4 simprr 499 . . . . . . . . . 10 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → (*Q𝑧) ∈ (1st𝐴))
5 simplr 497 . . . . . . . . . 10 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → (*Q𝑦) ∈ (2nd𝐴))
64, 5jca 300 . . . . . . . . 9 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → ((*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)))
7 prop 6727 . . . . . . . . . . 11 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
8 prltlu 6739 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → (*Q𝑧) <Q (*Q𝑦))
97, 8syl3an1 1203 . . . . . . . . . 10 ((𝐴P ∧ (*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → (*Q𝑧) <Q (*Q𝑦))
1093expb 1140 . . . . . . . . 9 ((𝐴P ∧ ((*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴))) → (*Q𝑧) <Q (*Q𝑦))
116, 10sylan2 280 . . . . . . . 8 ((𝐴P ∧ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴)))) → (*Q𝑧) <Q (*Q𝑦))
12 simprl 498 . . . . . . . . . . 11 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → 𝑧 <Q 𝑞)
13 simpll 496 . . . . . . . . . . 11 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → 𝑞 <Q 𝑦)
141, 2sotri 4750 . . . . . . . . . . 11 ((𝑧 <Q 𝑞𝑞 <Q 𝑦) → 𝑧 <Q 𝑦)
1512, 13, 14syl2anc 403 . . . . . . . . . 10 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
16 ltrnqi 6673 . . . . . . . . . 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 271 . . . . . . . 8 ((𝐴P ∧ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴)))) → (*Q𝑦) <Q (*Q𝑧))
1911, 18jca 300 . . . . . . 7 ((𝐴P ∧ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴)))) → ((*Q𝑧) <Q (*Q𝑦) ∧ (*Q𝑦) <Q (*Q𝑧)))
2019ex 113 . . . . . 6 (𝐴P → (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → ((*Q𝑧) <Q (*Q𝑦) ∧ (*Q𝑦) <Q (*Q𝑧))))
2120adantr 270 . . . . 5 ((𝐴P𝑞Q) → (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → ((*Q𝑧) <Q (*Q𝑦) ∧ (*Q𝑦) <Q (*Q𝑧))))
223, 21mtoi 623 . . . 4 ((𝐴P𝑞Q) → ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
2322alrimivv 1797 . . 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 6874 . . . . . . . 8 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
2624recexprlemelu 6875 . . . . . . . 8 (𝑞 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)))
2725, 26anbi12i 448 . . . . . . 7 ((𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴))))
28 breq1 3796 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑦 <Q 𝑞𝑧 <Q 𝑞))
29 fveq2 5209 . . . . . . . . . . 11 (𝑦 = 𝑧 → (*Q𝑦) = (*Q𝑧))
3029eleq1d 2148 . . . . . . . . . 10 (𝑦 = 𝑧 → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q𝑧) ∈ (1st𝐴)))
3128, 30anbi12d 457 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3231cbvexv 1837 . . . . . . . 8 (∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ ∃𝑧(𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴)))
3332anbi2i 445 . . . . . . 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 182 . . . . . 6 ((𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ ∃𝑧(𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
35 eeanv 1849 . . . . . 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 185 . . . . 5 ((𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ ∃𝑦𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3736notbii 627 . . . 4 (¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ ¬ ∃𝑦𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
38 alnex 1429 . . . . . 6 (∀𝑧 ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) ↔ ¬ ∃𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3938albii 1400 . . . . 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 1429 . . . . 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 182 . . . 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 185 . . 3 (¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ ∀𝑦𝑧 ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
4323, 42sylibr 132 . 2 ((𝐴P𝑞Q) → ¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)))
4443ralrimiva 2435 1 (𝐴P → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1283   = wceq 1285  wex 1422  wcel 1434  {cab 2068  wral 2349  cop 3409   class class class wbr 3793  cfv 4932  1st c1st 5796  2nd c2nd 5797  Qcnq 6532  *Qcrq 6536   <Q cltq 6537  Pcnp 6543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-mi 6558  df-lti 6559  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-inp 6718
This theorem is referenced by:  recexprlempr  6884
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