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Theorem inssdif 3224
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 3113 . . . 4 (𝑥𝐵 → ¬ 𝑥 ∈ (V ∖ 𝐵))
21anim2i 334 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
3 elin 3172 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
4 eldif 2997 . . 3 (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
52, 3, 43imtr4i 199 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
65ssriv 3018 1 (𝐴𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wcel 1436  Vcvv 2615  cdif 2985  cin 2987  wss 2988
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-dif 2990  df-in 2994  df-ss 3001
This theorem is referenced by:  difdif2ss  3245
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