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Theorem difdifdirss 3328
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))

Proof of Theorem difdifdirss
StepHypRef Expression
1 dif32 3228 . . . . 5 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)
2 invdif 3207 . . . . 5 ((𝐴𝐶) ∩ (V ∖ 𝐵)) = ((𝐴𝐶) ∖ 𝐵)
31, 2eqtr4i 2105 . . . 4 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (V ∖ 𝐵))
4 un0 3279 . . . 4 (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = ((𝐴𝐶) ∩ (V ∖ 𝐵))
53, 4eqtr4i 2105 . . 3 ((𝐴𝐵) ∖ 𝐶) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
6 indi 3212 . . . 4 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴𝐶) ∩ 𝐶))
7 disjdif 3317 . . . . . 6 (𝐶 ∩ (𝐴𝐶)) = ∅
8 incom 3159 . . . . . 6 (𝐶 ∩ (𝐴𝐶)) = ((𝐴𝐶) ∩ 𝐶)
97, 8eqtr3i 2104 . . . . 5 ∅ = ((𝐴𝐶) ∩ 𝐶)
109uneq2i 3124 . . . 4 (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴𝐶) ∩ 𝐶))
116, 10eqtr4i 2105 . . 3 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
125, 11eqtr4i 2105 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶))
13 ddifss 3203 . . . . . 6 𝐶 ⊆ (V ∖ (V ∖ 𝐶))
14 unss2 3144 . . . . . 6 (𝐶 ⊆ (V ∖ (V ∖ 𝐶)) → ((V ∖ 𝐵) ∪ 𝐶) ⊆ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))))
1513, 14ax-mp 7 . . . . 5 ((V ∖ 𝐵) ∪ 𝐶) ⊆ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶)))
16 indmss 3224 . . . . . 6 ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) ⊆ (V ∖ (𝐵 ∩ (V ∖ 𝐶)))
17 invdif 3207 . . . . . . 7 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
1817difeq2i 3088 . . . . . 6 (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = (V ∖ (𝐵𝐶))
1916, 18sseqtri 3032 . . . . 5 ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) ⊆ (V ∖ (𝐵𝐶))
2015, 19sstri 3009 . . . 4 ((V ∖ 𝐵) ∪ 𝐶) ⊆ (V ∖ (𝐵𝐶))
21 sslin 3193 . . . 4 (((V ∖ 𝐵) ∪ 𝐶) ⊆ (V ∖ (𝐵𝐶)) → ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) ⊆ ((𝐴𝐶) ∩ (V ∖ (𝐵𝐶))))
2220, 21ax-mp 7 . . 3 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) ⊆ ((𝐴𝐶) ∩ (V ∖ (𝐵𝐶)))
23 invdif 3207 . . 3 ((𝐴𝐶) ∩ (V ∖ (𝐵𝐶))) = ((𝐴𝐶) ∖ (𝐵𝐶))
2422, 23sseqtri 3032 . 2 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) ⊆ ((𝐴𝐶) ∖ (𝐵𝐶))
2512, 24eqsstri 3030 1 ((𝐴𝐵) ∖ 𝐶) ⊆ ((𝐴𝐶) ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  Vcvv 2602  cdif 2971  cun 2972  cin 2973  wss 2974  c0 3252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253
This theorem is referenced by: (None)
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