Theorem ddifss 3442

Description: Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3335), 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 3246	. 2 ⊢ 𝐴 ⊆ V
2		ssddif 3438	. 2 ⊢ (𝐴 ⊆ V ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐴)))
3	1, 2	mpbi 145	1 ⊢ 𝐴 ⊆ (V ∖ (V ∖ 𝐴))

Syntax hints:  Vcvv 2799   ∖ cdif 3194   ⊆ wss 3197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210

This theorem is referenced by:  ssindif0im  3551  difdifdirss  3576