Theorem ddifss 3345

Description: Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3238), 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 3150	. 2 |- A C_ _V
2		ssddif 3341	. 2 |- ( A C_ _V <-> A C_ ( _V \ ( _V \ A ) ) )
3	1, 2	mpbi 144	1 |- A C_ ( _V \ ( _V \ A ) )

Syntax hints:   _Vcvv 2712   \ cdif 3099   C_ wss 3102

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115

This theorem is referenced by:  ssindif0im  3453  difdifdirss  3478