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Theorem undifdcss 6804
 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 3147 . . . 4 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
2 undifss 3438 . . . 4 (𝐵𝐴 ↔ (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
31, 2sylibr 133 . . 3 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → 𝐵𝐴)
4 eleq2 2201 . . . . . . . 8 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → (𝑥𝐴𝑥 ∈ (𝐵 ∪ (𝐴𝐵))))
54biimpa 294 . . . . . . 7 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → 𝑥 ∈ (𝐵 ∪ (𝐴𝐵)))
6 elun 3212 . . . . . . 7 (𝑥 ∈ (𝐵 ∪ (𝐴𝐵)) ↔ (𝑥𝐵𝑥 ∈ (𝐴𝐵)))
75, 6sylib 121 . . . . . 6 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → (𝑥𝐵𝑥 ∈ (𝐴𝐵)))
8 eldifn 3194 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝐵)
98orim2i 750 . . . . . 6 ((𝑥𝐵𝑥 ∈ (𝐴𝐵)) → (𝑥𝐵 ∨ ¬ 𝑥𝐵))
107, 9syl 14 . . . . 5 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → (𝑥𝐵 ∨ ¬ 𝑥𝐵))
11 df-dc 820 . . . . 5 (DECID 𝑥𝐵 ↔ (𝑥𝐵 ∨ ¬ 𝑥𝐵))
1210, 11sylibr 133 . . . 4 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → DECID 𝑥𝐵)
1312ralrimiva 2503 . . 3 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → ∀𝑥𝐴 DECID 𝑥𝐵)
143, 13jca 304 . 2 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → (𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵))
15 elun1 3238 . . . . . . 7 (𝑦𝐵𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
1615adantl 275 . . . . . 6 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
17 simplr 519 . . . . . . . 8 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → 𝑦𝐴)
18 simpr 109 . . . . . . . 8 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → ¬ 𝑦𝐵)
1917, 18eldifd 3076 . . . . . . 7 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → 𝑦 ∈ (𝐴𝐵))
20 elun2 3239 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
2119, 20syl 14 . . . . . 6 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
22 eleq1 2200 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
2322dcbid 823 . . . . . . . 8 (𝑥 = 𝑦 → (DECID 𝑥𝐵DECID 𝑦𝐵))
24 simplr 519 . . . . . . . 8 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → ∀𝑥𝐴 DECID 𝑥𝐵)
25 simpr 109 . . . . . . . 8 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → 𝑦𝐴)
2623, 24, 25rspcdva 2789 . . . . . . 7 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → DECID 𝑦𝐵)
27 exmiddc 821 . . . . . . 7 (DECID 𝑦𝐵 → (𝑦𝐵 ∨ ¬ 𝑦𝐵))
2826, 27syl 14 . . . . . 6 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → (𝑦𝐵 ∨ ¬ 𝑦𝐵))
2916, 21, 28mpjaodan 787 . . . . 5 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
3029ex 114 . . . 4 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → (𝑦𝐴𝑦 ∈ (𝐵 ∪ (𝐴𝐵))))
3130ssrdv 3098 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → 𝐴 ⊆ (𝐵 ∪ (𝐴𝐵)))
322biimpi 119 . . . 4 (𝐵𝐴 → (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
3332adantr 274 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
3431, 33eqssd 3109 . 2 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → 𝐴 = (𝐵 ∪ (𝐴𝐵)))
3514, 34impbii 125 1 (𝐴 = (𝐵 ∪ (𝐴𝐵)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104   ∨ wo 697  DECID wdc 819   = wceq 1331   ∈ wcel 1480  ∀wral 2414   ∖ cdif 3063   ∪ cun 3064   ⊆ wss 3066 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079 This theorem is referenced by:  sbthlemi5  6842  sbthlemi6  6843  exmidfodomrlemim  7050
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