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Theorem inssdif 3339
 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 3227 . . . 4 (𝑥𝐵 → ¬ 𝑥 ∈ (V ∖ 𝐵))
21anim2i 340 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
3 elin 3286 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
4 eldif 3107 . . 3 (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
52, 3, 43imtr4i 200 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
65ssriv 3128 1 (𝐴𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 103   ∈ wcel 2125  Vcvv 2709   ∖ cdif 3095   ∩ cin 3097   ⊆ wss 3098 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-dif 3100  df-in 3104  df-ss 3111 This theorem is referenced by:  difdif2ss  3360
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