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Theorem difdif2ss 3303
Description: Set difference with a set difference. In classical logic this would be equality rather than subset. (Contributed by Jim Kingdon, 27-Jul-2018.)
Assertion
Ref Expression
difdif2ss ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))

Proof of Theorem difdif2ss
StepHypRef Expression
1 inssdif 3282 . . . 4 (𝐴𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶))
2 unss2 3217 . . . 4 ((𝐴𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶)) → ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))))
31, 2ax-mp 5 . . 3 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
4 difindiss 3300 . . 3 ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
53, 4sstri 3076 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
6 invdif 3288 . . . 4 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
76eqcomi 2121 . . 3 (𝐵𝐶) = (𝐵 ∩ (V ∖ 𝐶))
87difeq2i 3161 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
95, 8sseqtrri 3102 1 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  Vcvv 2660  cdif 3038  cun 3039  cin 3040  wss 3041
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054
This theorem is referenced by: (None)
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