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Theorem ssddif 3341
 Description: Double complement and subset. Similar to ddifss 3345 but inside a class instead of the universal class . In classical logic the subset operation on the right hand side could be an equality (that is, ). (Contributed by Jim Kingdon, 24-Jul-2018.)
Assertion
Ref Expression
ssddif

Proof of Theorem ssddif
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ancr 319 . . . . 5
2 simpr 109 . . . . . . . 8
32con2i 617 . . . . . . 7
43anim2i 340 . . . . . 6
5 eldif 3111 . . . . . . 7
6 eldif 3111 . . . . . . . . 9
76notbii 658 . . . . . . . 8
87anbi2i 453 . . . . . . 7
95, 8bitri 183 . . . . . 6
104, 9sylibr 133 . . . . 5
111, 10syl6 33 . . . 4
12 eldifi 3229 . . . . 5
1312imim2i 12 . . . 4
1411, 13impbii 125 . . 3
1514albii 1450 . 2
16 dfss2 3117 . 2
17 dfss2 3117 . 2
1815, 16, 173bitr4i 211 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wb 104  wal 1333   wcel 2128   cdif 3099   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:  ddifss  3345  inssddif  3348
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