Theorem ddifss 3411

Description: Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3304), 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 3215	. 2 |- A C_ _V
2		ssddif 3407	. 2 |- ( A C_ _V <-> A C_ ( _V \ ( _V \ A ) ) )
3	1, 2	mpbi 145	1 |- A C_ ( _V \ ( _V \ A ) )

Syntax hints:   _Vcvv 2772   \ cdif 3163   C_ wss 3166

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179

This theorem is referenced by:  ssindif0im  3520  difdifdirss  3545