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Theorem difindiss 3218
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 3111 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
2 eldif 2954 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
3 eldif 2954 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
42, 3orbi12i 691 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
5 andi 742 . . . . . 6 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
64, 5bitr4i 180 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)))
7 pm3.14 680 . . . . . 6 ((¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶) → ¬ (𝑥𝐵𝑥𝐶))
87anim2i 328 . . . . 5 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) → (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
96, 8sylbi 118 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) → (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
10 eldif 2954 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
11 elin 3153 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1211notbii 604 . . . . . 6 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
1312anbi2i 438 . . . . 5 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
1410, 13bitr2i 178 . . . 4 ((𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)) ↔ 𝑥 ∈ (𝐴 ∖ (𝐵𝐶)))
159, 14sylib 131 . . 3 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴 ∖ (𝐵𝐶)))
161, 15sylbi 118 . 2 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) → 𝑥 ∈ (𝐴 ∖ (𝐵𝐶)))
1716ssriv 2976 1 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101  wo 639  wcel 1409  cdif 2941  cun 2942  cin 2943  wss 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958
This theorem is referenced by:  difdif2ss  3221  indmss  3223
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