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Theorem difindiss 3191
 Description: Distributive law for class difference. In classical logic, for example, theorem 40 of [Suppes] p. 29, this is an equality instead of subset. (Contributed by Jim Kingdon, 26-Jul-2018.)
Assertion
Ref Expression
difindiss

Proof of Theorem difindiss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elun 3084 . . 3
2 eldif 2927 . . . . . . 7
3 eldif 2927 . . . . . . 7
42, 3orbi12i 681 . . . . . 6
5 andi 731 . . . . . 6
64, 5bitr4i 176 . . . . 5
7 pm3.14 670 . . . . . 6
87anim2i 324 . . . . 5
96, 8sylbi 114 . . . 4
10 eldif 2927 . . . . 5
11 elin 3126 . . . . . . 7
1211notbii 594 . . . . . 6
1312anbi2i 430 . . . . 5
1410, 13bitr2i 174 . . . 4
159, 14sylib 127 . . 3
161, 15sylbi 114 . 2
1716ssriv 2949 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 97   wo 629   wcel 1393   cdif 2914   cun 2915   cin 2916   wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931 This theorem is referenced by:  difdif2ss  3194  indmss  3196
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