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Theorem difdif2ss 3228
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 3207 . . . 4 (𝐴𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶))
2 unss2 3144 . . . 4 ((𝐴𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶)) → ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))))
31, 2ax-mp 7 . . 3 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
4 difindiss 3225 . . 3 ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
53, 4sstri 3009 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
6 invdif 3213 . . . 4 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
76eqcomi 2086 . . 3 (𝐵𝐶) = (𝐵 ∩ (V ∖ 𝐶))
87difeq2i 3088 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
95, 8sseqtr4i 3033 1 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  Vcvv 2602  cdif 2971  cun 2972  cin 2973  wss 2974
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
This theorem is referenced by: (None)
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