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 Description: Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression

StepHypRef Expression
1 dif32 3307 . . . . 5
2 invdif 3286 . . . . 5
31, 2eqtr4i 2139 . . . 4
4 un0 3364 . . . 4
53, 4eqtr4i 2139 . . 3
6 indi 3291 . . . 4
7 disjdif 3403 . . . . . 6
8 incom 3236 . . . . . 6
97, 8eqtr3i 2138 . . . . 5
109uneq2i 3195 . . . 4
116, 10eqtr4i 2139 . . 3
125, 11eqtr4i 2139 . 2
13 ddifss 3282 . . . . . 6
14 unss2 3215 . . . . . 6
1513, 14ax-mp 5 . . . . 5
16 indmss 3303 . . . . . 6
17 invdif 3286 . . . . . . 7
1817difeq2i 3159 . . . . . 6
1916, 18sseqtri 3099 . . . . 5
2015, 19sstri 3074 . . . 4
21 sslin 3270 . . . 4
2220, 21ax-mp 5 . . 3
23 invdif 3286 . . 3
2422, 23sseqtri 3099 . 2
2512, 24eqsstri 3097 1
 Colors of variables: wff set class Syntax hints:  cvv 2658   cdif 3036   cun 3037   cin 3038   wss 3039  c0 3331 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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332 This theorem is referenced by: (None)
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