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Theorem undifdcss 6713
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 3094 . . . 4 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
2 undifss 3382 . . . 4 (𝐵𝐴 ↔ (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
31, 2sylibr 133 . . 3 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → 𝐵𝐴)
4 eleq2 2158 . . . . . . . 8 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → (𝑥𝐴𝑥 ∈ (𝐵 ∪ (𝐴𝐵))))
54biimpa 291 . . . . . . 7 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → 𝑥 ∈ (𝐵 ∪ (𝐴𝐵)))
6 elun 3156 . . . . . . 7 (𝑥 ∈ (𝐵 ∪ (𝐴𝐵)) ↔ (𝑥𝐵𝑥 ∈ (𝐴𝐵)))
75, 6sylib 121 . . . . . 6 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → (𝑥𝐵𝑥 ∈ (𝐴𝐵)))
8 eldifn 3138 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝐵)
98orim2i 716 . . . . . 6 ((𝑥𝐵𝑥 ∈ (𝐴𝐵)) → (𝑥𝐵 ∨ ¬ 𝑥𝐵))
107, 9syl 14 . . . . 5 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → (𝑥𝐵 ∨ ¬ 𝑥𝐵))
11 df-dc 784 . . . . 5 (DECID 𝑥𝐵 ↔ (𝑥𝐵 ∨ ¬ 𝑥𝐵))
1210, 11sylibr 133 . . . 4 ((𝐴 = (𝐵 ∪ (𝐴𝐵)) ∧ 𝑥𝐴) → DECID 𝑥𝐵)
1312ralrimiva 2458 . . 3 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → ∀𝑥𝐴 DECID 𝑥𝐵)
143, 13jca 301 . 2 (𝐴 = (𝐵 ∪ (𝐴𝐵)) → (𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵))
15 elun1 3182 . . . . . . 7 (𝑦𝐵𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
1615adantl 272 . . . . . 6 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
17 simplr 498 . . . . . . . 8 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → 𝑦𝐴)
18 simpr 109 . . . . . . . 8 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → ¬ 𝑦𝐵)
1917, 18eldifd 3023 . . . . . . 7 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → 𝑦 ∈ (𝐴𝐵))
20 elun2 3183 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
2119, 20syl 14 . . . . . 6 ((((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
22 eleq1 2157 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
2322dcbid 789 . . . . . . . 8 (𝑥 = 𝑦 → (DECID 𝑥𝐵DECID 𝑦𝐵))
24 simplr 498 . . . . . . . 8 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → ∀𝑥𝐴 DECID 𝑥𝐵)
25 simpr 109 . . . . . . . 8 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → 𝑦𝐴)
2623, 24, 25rspcdva 2741 . . . . . . 7 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → DECID 𝑦𝐵)
27 exmiddc 785 . . . . . . 7 (DECID 𝑦𝐵 → (𝑦𝐵 ∨ ¬ 𝑦𝐵))
2826, 27syl 14 . . . . . 6 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → (𝑦𝐵 ∨ ¬ 𝑦𝐵))
2916, 21, 28mpjaodan 750 . . . . 5 (((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) ∧ 𝑦𝐴) → 𝑦 ∈ (𝐵 ∪ (𝐴𝐵)))
3029ex 114 . . . 4 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → (𝑦𝐴𝑦 ∈ (𝐵 ∪ (𝐴𝐵))))
3130ssrdv 3045 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → 𝐴 ⊆ (𝐵 ∪ (𝐴𝐵)))
322biimpi 119 . . . 4 (𝐵𝐴 → (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
3332adantr 271 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → (𝐵 ∪ (𝐴𝐵)) ⊆ 𝐴)
3431, 33eqssd 3056 . 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 667  DECID wdc 783   = wceq 1296  wcel 1445  wral 2370  cdif 3010  cun 3011  wss 3013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-dc 784  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026
This theorem is referenced by:  sbthlemi5  6750  sbthlemi6  6751  exmidfodomrlemim  6924
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