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Theorem unssdif 3281
Description: Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
Assertion
Ref Expression
unssdif (𝐴𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∖ 𝐵))

Proof of Theorem unssdif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 2663 . . . . . . . 8 𝑥 ∈ V
2 eldif 3050 . . . . . . . 8 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
31, 2mpbiran 909 . . . . . . 7 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
43anbi1i 453 . . . . . 6 ((𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥𝐵) ↔ (¬ 𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 eldif 3050 . . . . . 6 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥𝐵))
6 ioran 726 . . . . . 6 (¬ (𝑥𝐴𝑥𝐵) ↔ (¬ 𝑥𝐴 ∧ ¬ 𝑥𝐵))
74, 5, 63bitr4i 211 . . . . 5 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
87biimpi 119 . . . 4 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) → ¬ (𝑥𝐴𝑥𝐵))
98con2i 601 . . 3 ((𝑥𝐴𝑥𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
10 elun 3187 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
11 eldif 3050 . . . 4 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵)))
121, 11mpbiran 909 . . 3 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
139, 10, 123imtr4i 200 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)))
1413ssriv 3071 1 (𝐴𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∖ 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wo 682  wcel 1465  Vcvv 2660  cdif 3038  cun 3039  wss 3041
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054
This theorem is referenced by: (None)
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