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Theorem difindiss 3185
 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 3078 . . 3 (x ((AB) ∪ (A𝐶)) ↔ (x (AB) x (A𝐶)))
2 eldif 2921 . . . . . . 7 (x (AB) ↔ (x A ¬ x B))
3 eldif 2921 . . . . . . 7 (x (A𝐶) ↔ (x A ¬ x 𝐶))
42, 3orbi12i 680 . . . . . 6 ((x (AB) x (A𝐶)) ↔ ((x A ¬ x B) (x A ¬ x 𝐶)))
5 andi 730 . . . . . 6 ((x A x B ¬ x 𝐶)) ↔ ((x A ¬ x B) (x A ¬ x 𝐶)))
64, 5bitr4i 176 . . . . 5 ((x (AB) x (A𝐶)) ↔ (x A x B ¬ x 𝐶)))
7 pm3.14 669 . . . . . 6 ((¬ x B ¬ x 𝐶) → ¬ (x B x 𝐶))
87anim2i 324 . . . . 5 ((x A x B ¬ x 𝐶)) → (x A ¬ (x B x 𝐶)))
96, 8sylbi 114 . . . 4 ((x (AB) x (A𝐶)) → (x A ¬ (x B x 𝐶)))
10 eldif 2921 . . . . 5 (x (A ∖ (B𝐶)) ↔ (x A ¬ x (B𝐶)))
11 elin 3120 . . . . . . 7 (x (B𝐶) ↔ (x B x 𝐶))
1211notbii 593 . . . . . 6 x (B𝐶) ↔ ¬ (x B x 𝐶))
1312anbi2i 430 . . . . 5 ((x A ¬ x (B𝐶)) ↔ (x A ¬ (x B x 𝐶)))
1410, 13bitr2i 174 . . . 4 ((x A ¬ (x B x 𝐶)) ↔ 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 2943 1 ((AB) ∪ (A𝐶)) ⊆ (A ∖ (B𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ∨ wo 628   ∈ wcel 1390   ∖ cdif 2908   ∪ cun 2909   ∩ cin 2910   ⊆ wss 2911 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925 This theorem is referenced by:  difdif2ss  3188  indmss  3190
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