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

Theorem undifdcss 6560
Description: Union of complementary parts into whole and decidability. (Contributed by Jim Kingdon, 17-Jun-2022.)
Assertion
Ref Expression
undifdcss (𝐴 = (𝐵 ∪ (𝐴𝐵)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem undifdcss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqimss2 3063 . . . 4 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
2 undifss 3344 . . . 4 (𝐵𝐴 ↔ (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
31, 2sylibr 132 . . 3 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → 𝐵𝐴)
4 eleq2 2146 . . . . . . . 8 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → (𝑥𝐴𝑥 ∈ (𝐵 ∪ (𝐴𝐵))))
54biimpa 290 . . . . . . 7 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → 𝑥 ∈ (𝐵 ∪ (𝐴𝐵)))
6 elun 3125 . . . . . . 7 (𝑥 ∈ (𝐵 ∪ (𝐴𝐵)) ↔ (𝑥𝐵𝑥 ∈ (𝐴𝐵)))
75, 6sylib 120 . . . . . 6 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → (𝑥𝐵𝑥 ∈ (𝐴𝐵)))
8 eldifn 3107 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝐵)
98orim2i 711 . . . . . 6 ((𝑥𝐵𝑥 ∈ (𝐴𝐵)) → (𝑥𝐵 ∨ ¬ 𝑥𝐵))
107, 9syl 14 . . . . 5 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → (𝑥𝐵 ∨ ¬ 𝑥𝐵))
11 df-dc 777 . . . . 5 (DECID 𝑥𝐵 ↔ (𝑥𝐵 ∨ ¬ 𝑥𝐵))
1210, 11sylibr 132 . . . 4 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → DECID 𝑥𝐵)
1312ralrimiva 2440 . . 3 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → ∀𝑥𝐴 DECID 𝑥𝐵)
143, 13jca 300 . 2 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → (𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵))
15 elun1 3151 . . . . . . 7 (𝑦𝐵𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
1615adantl 271 . . . . . 6 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
17 simplr 497 . . . . . . . 8 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → 𝑦𝐴)
18 simpr 108 . . . . . . . 8 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → ¬ 𝑦𝐵)
1917, 18eldifd 2994 . . . . . . 7 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → 𝑦 ∈ (𝐴𝐵))
20 elun2 3152 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
2119, 20syl 14 . . . . . 6 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
22 eleq1 2145 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
2322dcbid 782 . . . . . . . 8 (𝑥 = 𝑦 → (DECID 𝑥𝐵DECID 𝑦𝐵))
24 simplr 497 . . . . . . . 8 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → ∀𝑥𝐴 DECID 𝑥𝐵)
25 simpr 108 . . . . . . . 8 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → 𝑦𝐴)
2623, 24, 25rspcdva 2717 . . . . . . 7 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → DECID 𝑦𝐵)
27 exmiddc 778 . . . . . . 7 (DECID 𝑦𝐵 → (𝑦𝐵 ∨ ¬ 𝑦𝐵))
2826, 27syl 14 . . . . . 6 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → (𝑦𝐵 ∨ ¬ 𝑦𝐵))
2916, 21, 28mpjaodan 745 . . . . 5 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
3029ex 113 . . . 4 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → (𝑦𝐴𝑦 ∈ (𝐵 ∪ (𝐴𝐵))))
3130ssrdv 3016 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → 𝐴 ⊆ (𝐵 ∪ (𝐴𝐵)))
322biimpi 118 . . . 4 (𝐵𝐴 → (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
3332adantr 270 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
3431, 33eqssd 3027 . 2 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → 𝐴 = (𝐵 ∪ (𝐴𝐵)))
3514, 34impbii 124 1 (𝐴 = (𝐵 ∪ (𝐴𝐵)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wb 103  wo 662  DECID wdc 776   = wceq 1285  wcel 1434  wral 2353  cdif 2981  cun 2982  wss 2984
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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997
This theorem is referenced by:  exmidfodomrlemim  6730
  Copyright terms: Public domain W3C validator