Theorem ddifss 3202

Description: Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3103), 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

Step	Hyp	Ref	Expression
1		ssv 3019	. 2 ⊢ 𝐴 ⊆ V
2		ssddif 3198	. 2 ⊢ (𝐴 ⊆ V ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐴)))
3	1, 2	mpbi 143	1 ⊢ 𝐴 ⊆ (V ∖ (V ∖ 𝐴))

Syntax hints:  Vcvv 2601   ∖ cdif 2970   ⊆ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986

This theorem is referenced by:  ssindif0im  3303  difdifdirss  3327