ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recexprlemdisj GIF version

Theorem recexprlemdisj 7402
Description: 𝐵 is disjoint. Lemma for recexpr 7410. (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 7170 . . . . . 6 <Q Or Q
2 ltrelnq 7137 . . . . . 6 <Q ⊆ (Q × Q)
31, 2son2lpi 4903 . . . . 5 ¬ ((*Q𝑧) <Q (*Q𝑦) ∧ (*Q𝑦) <Q (*Q𝑧))
4 simprr 504 . . . . . . . . . 10 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → (*Q𝑧) ∈ (1st𝐴))
5 simplr 502 . . . . . . . . . 10 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → (*Q𝑦) ∈ (2nd𝐴))
64, 5jca 302 . . . . . . . . 9 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → ((*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)))
7 prop 7247 . . . . . . . . . . 11 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
8 prltlu 7259 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → (*Q𝑧) <Q (*Q𝑦))
97, 8syl3an1 1232 . . . . . . . . . 10 ((𝐴P ∧ (*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → (*Q𝑧) <Q (*Q𝑦))
1093expb 1165 . . . . . . . . 9 ((𝐴P ∧ ((*Q𝑧) ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴))) → (*Q𝑧) <Q (*Q𝑦))
116, 10sylan2 282 . . . . . . . 8 ((𝐴P ∧ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴)))) → (*Q𝑧) <Q (*Q𝑦))
12 simprl 503 . . . . . . . . . . 11 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → 𝑧 <Q 𝑞)
13 simpll 501 . . . . . . . . . . 11 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → 𝑞 <Q 𝑦)
141, 2sotri 4902 . . . . . . . . . . 11 ((𝑧 <Q 𝑞𝑞 <Q 𝑦) → 𝑧 <Q 𝑦)
1512, 13, 14syl2anc 406 . . . . . . . . . 10 (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
16 ltrnqi 7193 . . . . . . . . . 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 273 . . . . . . . 8 ((𝐴P ∧ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴)))) → (*Q𝑦) <Q (*Q𝑧))
1911, 18jca 302 . . . . . . 7 ((𝐴P ∧ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴)))) → ((*Q𝑧) <Q (*Q𝑦) ∧ (*Q𝑦) <Q (*Q𝑧)))
2019ex 114 . . . . . 6 (𝐴P → (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → ((*Q𝑧) <Q (*Q𝑦) ∧ (*Q𝑦) <Q (*Q𝑧))))
2120adantr 272 . . . . 5 ((𝐴P𝑞Q) → (((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) → ((*Q𝑧) <Q (*Q𝑦) ∧ (*Q𝑦) <Q (*Q𝑧))))
223, 21mtoi 636 . . . 4 ((𝐴P𝑞Q) → ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
2322alrimivv 1829 . . 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 7394 . . . . . . . 8 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
2624recexprlemelu 7395 . . . . . . . 8 (𝑞 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)))
2725, 26anbi12i 453 . . . . . . 7 ((𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴))))
28 breq1 3900 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑦 <Q 𝑞𝑧 <Q 𝑞))
29 fveq2 5387 . . . . . . . . . . 11 (𝑦 = 𝑧 → (*Q𝑦) = (*Q𝑧))
3029eleq1d 2184 . . . . . . . . . 10 (𝑦 = 𝑧 → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q𝑧) ∈ (1st𝐴)))
3128, 30anbi12d 462 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3231cbvexv 1870 . . . . . . . 8 (∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ ∃𝑧(𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴)))
3332anbi2i 450 . . . . . . 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 183 . . . . . 6 ((𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ ∃𝑧(𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
35 eeanv 1882 . . . . . 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 186 . . . . 5 ((𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ ∃𝑦𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3736notbii 640 . . . 4 (¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ ¬ ∃𝑦𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
38 alnex 1458 . . . . . 6 (∀𝑧 ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))) ↔ ¬ ∃𝑧((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
3938albii 1429 . . . . 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 1458 . . . . 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 183 . . . 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 186 . . 3 (¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ↔ ∀𝑦𝑧 ¬ ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ∧ (𝑧 <Q 𝑞 ∧ (*Q𝑧) ∈ (1st𝐴))))
4323, 42sylibr 133 . 2 ((𝐴P𝑞Q) → ¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)))
4443ralrimiva 2480 1 (𝐴P → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wal 1312   = wceq 1314  wex 1451  wcel 1463  {cab 2101  wral 2391  cop 3498   class class class wbr 3897  cfv 5091  1st c1st 6002  2nd c2nd 6003  Qcnq 7052  *Qcrq 7056   <Q cltq 7057  Pcnp 7063
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-mi 7078  df-lti 7079  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-inp 7238
This theorem is referenced by:  recexprlempr  7404
  Copyright terms: Public domain W3C validator