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Theorem difindiss 3372
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 3259 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
2 eldif 3121 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
3 eldif 3121 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
42, 3orbi12i 754 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
5 andi 808 . . . . . 6 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
64, 5bitr4i 186 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)))
7 pm3.14 743 . . . . . 6 ((¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶) → ¬ (𝑥𝐵𝑥𝐶))
87anim2i 340 . . . . 5 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) → (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
96, 8sylbi 120 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) → (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
10 eldif 3121 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
11 elin 3301 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1211notbii 658 . . . . . 6 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
1312anbi2i 453 . . . . 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 3142 1 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wo 698  wcel 2135  cdif 3109  cun 3110  cin 3111  wss 3112
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125
This theorem is referenced by:  difdif2ss  3375  indmss  3377
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