Theorem ddifss 3314

Description: Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3207), 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 3119	. 2 |- A C_ _V
2		ssddif 3310	. 2 |- ( A C_ _V <-> A C_ ( _V \ ( _V \ A ) ) )
3	1, 2	mpbi 144	1 |- A C_ ( _V \ ( _V \ A ) )

Syntax hints:   _Vcvv 2686   \ cdif 3068   C_ wss 3071

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084

This theorem is referenced by:  ssindif0im  3422  difdifdirss  3447