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Theorem difindiss 3401
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 3288 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
2 eldif 3150 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
3 eldif 3150 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
42, 3orbi12i 765 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
5 andi 819 . . . . . 6 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
64, 5bitr4i 187 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)))
7 pm3.14 754 . . . . . 6 ((¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶) → ¬ (𝑥𝐵𝑥𝐶))
87anim2i 342 . . . . 5 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) → (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
96, 8sylbi 121 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) → (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
10 eldif 3150 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
11 elin 3330 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1211notbii 669 . . . . . 6 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
1312anbi2i 457 . . . . 5 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
1410, 13bitr2i 185 . . . 4 ((𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)) ↔ 𝑥 ∈ (𝐴 ∖ (𝐵𝐶)))
159, 14sylib 122 . . 3 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴 ∖ (𝐵𝐶)))
161, 15sylbi 121 . 2 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) → 𝑥 ∈ (𝐴 ∖ (𝐵𝐶)))
1716ssriv 3171 1 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wo 709  wcel 2158  cdif 3138  cun 3139  cin 3140  wss 3141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154
This theorem is referenced by:  difdif2ss  3404  indmss  3406
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