Theorem ddifss 3388

Description: Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3281), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)

Assertion

Ref	Expression
ddifss	|- A C_ ( _V \ ( _V \ A ) )

Proof of Theorem ddifss

Step	Hyp	Ref	Expression
1		ssv 3192	. 2 |- A C_ _V
2		ssddif 3384	. 2 |- ( A C_ _V <-> A C_ ( _V \ ( _V \ A ) ) )
3	1, 2	mpbi 145	1 |- A C_ ( _V \ ( _V \ A ) )

Syntax hints:   _Vcvv 2752   \ cdif 3141   C_ wss 3144

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157

This theorem is referenced by:  ssindif0im  3497  difdifdirss  3522