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Theorem inssdif 3278
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 3166 . . . 4 (𝑥𝐵 → ¬ 𝑥 ∈ (V ∖ 𝐵))
21anim2i 337 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
3 elin 3225 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
4 eldif 3046 . . 3 (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
52, 3, 43imtr4i 200 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
65ssriv 3067 1 (𝐴𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wcel 1463  Vcvv 2657  cdif 3034  cin 3036  wss 3037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-dif 3039  df-in 3043  df-ss 3050
This theorem is referenced by:  difdif2ss  3299
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