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Theorem difdifdirss 3367
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 3262 . . . . 5 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)
2 invdif 3241 . . . . 5 ((𝐴𝐶) ∩ (V ∖ 𝐵)) = ((𝐴𝐶) ∖ 𝐵)
31, 2eqtr4i 2111 . . . 4 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (V ∖ 𝐵))
4 un0 3316 . . . 4 (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = ((𝐴𝐶) ∩ (V ∖ 𝐵))
53, 4eqtr4i 2111 . . 3 ((𝐴𝐵) ∖ 𝐶) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
6 indi 3246 . . . 4 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴𝐶) ∩ 𝐶))
7 disjdif 3355 . . . . . 6 (𝐶 ∩ (𝐴𝐶)) = ∅
8 incom 3192 . . . . . 6 (𝐶 ∩ (𝐴𝐶)) = ((𝐴𝐶) ∩ 𝐶)
97, 8eqtr3i 2110 . . . . 5 ∅ = ((𝐴𝐶) ∩ 𝐶)
109uneq2i 3151 . . . 4 (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴𝐶) ∩ 𝐶))
116, 10eqtr4i 2111 . . 3 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
125, 11eqtr4i 2111 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶))
13 ddifss 3237 . . . . . 6 𝐶 ⊆ (V ∖ (V ∖ 𝐶))
14 unss2 3171 . . . . . 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 3258 . . . . . 6 ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) ⊆ (V ∖ (𝐵 ∩ (V ∖ 𝐶)))
17 invdif 3241 . . . . . . 7 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
1817difeq2i 3115 . . . . . 6 (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = (V ∖ (𝐵𝐶))
1916, 18sseqtri 3058 . . . . 5 ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) ⊆ (V ∖ (𝐵𝐶))
2015, 19sstri 3034 . . . 4 ((V ∖ 𝐵) ∪ 𝐶) ⊆ (V ∖ (𝐵𝐶))
21 sslin 3226 . . . 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 3241 . . 3 ((𝐴𝐶) ∩ (V ∖ (𝐵𝐶))) = ((𝐴𝐶) ∖ (𝐵𝐶))
2422, 23sseqtri 3058 . 2 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) ⊆ ((𝐴𝐶) ∖ (𝐵𝐶))
2512, 24eqsstri 3056 1 ((𝐴𝐵) ∖ 𝐶) ⊆ ((𝐴𝐶) ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  Vcvv 2619  cdif 2996  cun 2997  cin 2998  wss 2999  c0 3286
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287
This theorem is referenced by: (None)
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