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Theorem difindiss 3330
 Description: Distributive law for class difference. In classical logic, for example, theorem 40 of [Suppes] p. 29, this is an equality instead of subset. (Contributed by Jim Kingdon, 26-Jul-2018.)
Assertion
Ref Expression
difindiss ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))

Proof of Theorem difindiss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elun 3217 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
2 eldif 3080 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
3 eldif 3080 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
42, 3orbi12i 753 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
5 andi 807 . . . . . 6 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
64, 5bitr4i 186 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)))
7 pm3.14 742 . . . . . 6 ((¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶) → ¬ (𝑥𝐵𝑥𝐶))
87anim2i 339 . . . . 5 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) → (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
96, 8sylbi 120 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) → (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
10 eldif 3080 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
11 elin 3259 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1211notbii 657 . . . . . 6 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
1312anbi2i 452 . . . . 5 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
1410, 13bitr2i 184 . . . 4 ((𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)) ↔ 𝑥 ∈ (𝐴 ∖ (𝐵𝐶)))
159, 14sylib 121 . . 3 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴 ∖ (𝐵𝐶)))
161, 15sylbi 120 . 2 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) → 𝑥 ∈ (𝐴 ∖ (𝐵𝐶)))
1716ssriv 3101 1 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 697   ∈ wcel 1480   ∖ cdif 3068   ∪ cun 3069   ∩ cin 3070   ⊆ wss 3071 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 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084 This theorem is referenced by:  difdif2ss  3333  indmss  3335
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