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Theorem difindiss 3188
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 3081 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
2 eldif 2924 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
3 eldif 2924 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
42, 3orbi12i 681 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
5 andi 731 . . . . . 6 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
64, 5bitr4i 176 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)))
7 pm3.14 670 . . . . . 6 ((¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶) → ¬ (𝑥𝐵𝑥𝐶))
87anim2i 324 . . . . 5 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) → (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
96, 8sylbi 114 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) → (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
10 eldif 2924 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
11 elin 3123 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1211notbii 594 . . . . . 6 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
1312anbi2i 430 . . . . 5 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
1410, 13bitr2i 174 . . . 4 ((𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)) ↔ 𝑥 ∈ (𝐴 ∖ (𝐵𝐶)))
159, 14sylib 127 . . 3 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴 ∖ (𝐵𝐶)))
161, 15sylbi 114 . 2 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) → 𝑥 ∈ (𝐴 ∖ (𝐵𝐶)))
1716ssriv 2946 1 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 97  wo 629  wcel 1393  cdif 2911  cun 2912  cin 2913  wss 2914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928
This theorem is referenced by:  difdif2ss  3191  indmss  3193
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