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Theorem ddifss 3284
Description: Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3177), 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 3089 . 2 𝐴 ⊆ V
2 ssddif 3280 . 2 (𝐴 ⊆ V ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐴)))
31, 2mpbi 144 1 𝐴 ⊆ (V ∖ (V ∖ 𝐴))
Colors of variables: wff set class
Syntax hints:  Vcvv 2660  cdif 3038  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-in 3047  df-ss 3054
This theorem is referenced by:  ssindif0im  3392  difdifdirss  3417
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