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Theorem inssdif 3170
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 3065 . . . 4 (𝑥𝐵 → ¬ 𝑥 ∈ (V ∖ 𝐵))
21anim2i 324 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
3 elin 3123 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
4 eldif 2924 . . 3 (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
52, 3, 43imtr4i 190 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
65ssriv 2946 1 (𝐴𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 97  wcel 1393  Vcvv 2554  cdif 2911  cin 2913  wss 2914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-dif 2917  df-in 2921  df-ss 2928
This theorem is referenced by:  difdif2ss  3191
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