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 Description: Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
difdifdirss ((𝐴𝐵) ∖ 𝐶) ⊆ ((𝐴𝐶) ∖ (𝐵𝐶))

StepHypRef Expression
1 dif32 3339 . . . . 5 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)
2 invdif 3318 . . . . 5 ((𝐴𝐶) ∩ (V ∖ 𝐵)) = ((𝐴𝐶) ∖ 𝐵)
31, 2eqtr4i 2163 . . . 4 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (V ∖ 𝐵))
4 un0 3396 . . . 4 (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = ((𝐴𝐶) ∩ (V ∖ 𝐵))
53, 4eqtr4i 2163 . . 3 ((𝐴𝐵) ∖ 𝐶) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
6 indi 3323 . . . 4 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴𝐶) ∩ 𝐶))
7 disjdif 3435 . . . . . 6 (𝐶 ∩ (𝐴𝐶)) = ∅
8 incom 3268 . . . . . 6 (𝐶 ∩ (𝐴𝐶)) = ((𝐴𝐶) ∩ 𝐶)
97, 8eqtr3i 2162 . . . . 5 ∅ = ((𝐴𝐶) ∩ 𝐶)
109uneq2i 3227 . . . 4 (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴𝐶) ∩ 𝐶))
116, 10eqtr4i 2163 . . 3 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
125, 11eqtr4i 2163 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶))
13 ddifss 3314 . . . . . 6 𝐶 ⊆ (V ∖ (V ∖ 𝐶))
14 unss2 3247 . . . . . 6 (𝐶 ⊆ (V ∖ (V ∖ 𝐶)) → ((V ∖ 𝐵) ∪ 𝐶) ⊆ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))))
1513, 14ax-mp 5 . . . . 5 ((V ∖ 𝐵) ∪ 𝐶) ⊆ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶)))
16 indmss 3335 . . . . . 6 ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) ⊆ (V ∖ (𝐵 ∩ (V ∖ 𝐶)))
17 invdif 3318 . . . . . . 7 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
1817difeq2i 3191 . . . . . 6 (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = (V ∖ (𝐵𝐶))
1916, 18sseqtri 3131 . . . . 5 ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) ⊆ (V ∖ (𝐵𝐶))
2015, 19sstri 3106 . . . 4 ((V ∖ 𝐵) ∪ 𝐶) ⊆ (V ∖ (𝐵𝐶))
21 sslin 3302 . . . 4 (((V ∖ 𝐵) ∪ 𝐶) ⊆ (V ∖ (𝐵𝐶)) → ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) ⊆ ((𝐴𝐶) ∩ (V ∖ (𝐵𝐶))))
2220, 21ax-mp 5 . . 3 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) ⊆ ((𝐴𝐶) ∩ (V ∖ (𝐵𝐶)))
23 invdif 3318 . . 3 ((𝐴𝐶) ∩ (V ∖ (𝐵𝐶))) = ((𝐴𝐶) ∖ (𝐵𝐶))
2422, 23sseqtri 3131 . 2 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) ⊆ ((𝐴𝐶) ∖ (𝐵𝐶))
2512, 24eqsstri 3129 1 ((𝐴𝐵) ∖ 𝐶) ⊆ ((𝐴𝐶) ∖ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:  Vcvv 2686   ∖ cdif 3068   ∪ cun 3069   ∩ cin 3070   ⊆ wss 3071  ∅c0 3363 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364 This theorem is referenced by: (None)
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