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Theorem ddifss 3314
 Description: Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3207), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
Assertion
Ref Expression
ddifss 𝐴 ⊆ (V ∖ (V ∖ 𝐴))

Proof of Theorem ddifss
StepHypRef Expression
1 ssv 3119 . 2 𝐴 ⊆ V
2 ssddif 3310 . 2 (𝐴 ⊆ V ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐴)))
31, 2mpbi 144 1 𝐴 ⊆ (V ∖ (V ∖ 𝐴))
 Colors of variables: wff set class Syntax hints:  Vcvv 2686   ∖ cdif 3068   ⊆ wss 3071 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084 This theorem is referenced by:  ssindif0im  3422  difdifdirss  3447
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