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Theorem recexprlemdisj 7168
Description: 𝐵 is disjoint. Lemma for recexpr 7176. (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 6936 . . . . . 6 <Q Or Q
2 ltrelnq 6903 . . . . . 6 <Q ⊆ (Q × Q)
31, 2son2lpi 4815 . . . . 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 7013 . . . . . . . . . . 11 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
8 prltlu 7025 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → (*Q𝑧) <Q (*Q𝑦))
97, 8syl3an1 1207 . . . . . . . . . 10 ((𝐴P ∧ (*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → (*Q𝑧) <Q (*Q𝑦))
1093expb 1144 . . . . . . . . 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 4814 . . . . . . . . . . 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 6959 . . . . . . . . . 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 625 . . . 4 ((𝐴P𝑞Q) → ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
2322alrimivv 1803 . . 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 7160 . . . . . . . 8 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
2624recexprlemelu 7161 . . . . . . . 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 3840 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑦 <Q 𝑞𝑧 <Q 𝑞))
29 fveq2 5289 . . . . . . . . . . 11 (𝑦 = 𝑧 → (*Q𝑦) = (*Q𝑧))
3029eleq1d 2156 . . . . . . . . . 10 (𝑦 = 𝑧 → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q𝑧) ∈ (1st𝐴)))
3128, 30anbi12d 457 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3231cbvexv 1843 . . . . . . . 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 1855 . . . . . 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 629 . . . 4 (¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ ¬ ∃𝑦𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
38 alnex 1433 . . . . . 6 (∀𝑧 ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) ↔ ¬ ∃𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3938albii 1404 . . . . 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 1433 . . . . 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 2446 1 (𝐴P → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1287   = wceq 1289  wex 1426  wcel 1438  {cab 2074  wral 2359  cop 3444   class class class wbr 3837  cfv 5002  1st c1st 5891  2nd c2nd 5892  Qcnq 6818  *Qcrq 6822   <Q cltq 6823  Pcnp 6829
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-mi 6844  df-lti 6845  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-inp 7004
This theorem is referenced by:  recexprlempr  7170
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