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Theorem unssdif 3317
 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 2693 . . . . . . . 8 𝑥 ∈ V
2 eldif 3086 . . . . . . . 8 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
31, 2mpbiran 925 . . . . . . 7 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
43anbi1i 454 . . . . . 6 ((𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥𝐵) ↔ (¬ 𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 eldif 3086 . . . . . 6 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥𝐵))
6 ioran 742 . . . . . 6 (¬ (𝑥𝐴𝑥𝐵) ↔ (¬ 𝑥𝐴 ∧ ¬ 𝑥𝐵))
74, 5, 63bitr4i 211 . . . . 5 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
87biimpi 119 . . . 4 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) → ¬ (𝑥𝐴𝑥𝐵))
98con2i 617 . . 3 ((𝑥𝐴𝑥𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
10 elun 3223 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
11 eldif 3086 . . . 4 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵)))
121, 11mpbiran 925 . . 3 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
139, 10, 123imtr4i 200 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)))
1413ssriv 3107 1 (𝐴𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∖ 𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 698   ∈ wcel 1481  Vcvv 2690   ∖ cdif 3074   ∪ cun 3075   ⊆ wss 3077 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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090 This theorem is referenced by: (None)
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