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Theorem inssdif 3386
Description: Intersection of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
Assertion
Ref Expression
inssdif (𝐴𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵))

Proof of Theorem inssdif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elndif 3274 . . . 4 (𝑥𝐵 → ¬ 𝑥 ∈ (V ∖ 𝐵))
21anim2i 342 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
3 elin 3333 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
4 eldif 3153 . . 3 (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
52, 3, 43imtr4i 201 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
65ssriv 3174 1 (𝐴𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wcel 2160  Vcvv 2752  cdif 3141  cin 3143  wss 3144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157
This theorem is referenced by:  difdif2ss  3407
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