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

Theorem undifdcss 6925
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 3212 . . . 4 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
2 undifss 3505 . . . 4 (𝐵𝐴 ↔ (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
31, 2sylibr 134 . . 3 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → 𝐵𝐴)
4 eleq2 2241 . . . . . . . 8 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → (𝑥𝐴𝑥 ∈ (𝐵 ∪ (𝐴𝐵))))
54biimpa 296 . . . . . . 7 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → 𝑥 ∈ (𝐵 ∪ (𝐴𝐵)))
6 elun 3278 . . . . . . 7 (𝑥 ∈ (𝐵 ∪ (𝐴𝐵)) ↔ (𝑥𝐵𝑥 ∈ (𝐴𝐵)))
75, 6sylib 122 . . . . . 6 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → (𝑥𝐵𝑥 ∈ (𝐴𝐵)))
8 eldifn 3260 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝐵)
98orim2i 761 . . . . . 6 ((𝑥𝐵𝑥 ∈ (𝐴𝐵)) → (𝑥𝐵 ∨ ¬ 𝑥𝐵))
107, 9syl 14 . . . . 5 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → (𝑥𝐵 ∨ ¬ 𝑥𝐵))
11 df-dc 835 . . . . 5 (DECID 𝑥𝐵 ↔ (𝑥𝐵 ∨ ¬ 𝑥𝐵))
1210, 11sylibr 134 . . . 4 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → DECID 𝑥𝐵)
1312ralrimiva 2550 . . 3 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → ∀𝑥𝐴 DECID 𝑥𝐵)
143, 13jca 306 . 2 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → (𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵))
15 elun1 3304 . . . . . . 7 (𝑦𝐵𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
1615adantl 277 . . . . . 6 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
17 simplr 528 . . . . . . . 8 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → 𝑦𝐴)
18 simpr 110 . . . . . . . 8 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → ¬ 𝑦𝐵)
1917, 18eldifd 3141 . . . . . . 7 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → 𝑦 ∈ (𝐴𝐵))
20 elun2 3305 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
2119, 20syl 14 . . . . . 6 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
22 eleq1 2240 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
2322dcbid 838 . . . . . . . 8 (𝑥 = 𝑦 → (DECID 𝑥𝐵DECID 𝑦𝐵))
24 simplr 528 . . . . . . . 8 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → ∀𝑥𝐴 DECID 𝑥𝐵)
25 simpr 110 . . . . . . . 8 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → 𝑦𝐴)
2623, 24, 25rspcdva 2848 . . . . . . 7 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → DECID 𝑦𝐵)
27 exmiddc 836 . . . . . . 7 (DECID 𝑦𝐵 → (𝑦𝐵 ∨ ¬ 𝑦𝐵))
2826, 27syl 14 . . . . . 6 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → (𝑦𝐵 ∨ ¬ 𝑦𝐵))
2916, 21, 28mpjaodan 798 . . . . 5 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
3029ex 115 . . . 4 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → (𝑦𝐴𝑦 ∈ (𝐵 ∪ (𝐴𝐵))))
3130ssrdv 3163 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → 𝐴 ⊆ (𝐵 ∪ (𝐴𝐵)))
322biimpi 120 . . . 4 (𝐵𝐴 → (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
3332adantr 276 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
3431, 33eqssd 3174 . 2 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → 𝐴 = (𝐵 ∪ (𝐴𝐵)))
3514, 34impbii 126 1 (𝐴 = (𝐵 ∪ (𝐴𝐵)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wb 105  wo 708  DECID wdc 834   = wceq 1353  wcel 2148  wral 2455  cdif 3128  cun 3129  wss 3131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144
This theorem is referenced by:  sbthlemi5  6963  sbthlemi6  6964  exmidfodomrlemim  7203  bj-charfundcALT  14701
  Copyright terms: Public domain W3C validator