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Theorem difdifdir 3483
 Description: Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)
Assertion
Ref Expression
difdifdir

Proof of Theorem difdifdir
StepHypRef Expression
1 dif32 3373 . . . . 5
2 invdif 3352 . . . . 5
31, 2eqtr4i 2279 . . . 4
4 un0 3421 . . . 4
53, 4eqtr4i 2279 . . 3
6 indi 3357 . . . 4
7 disjdif 3468 . . . . . 6
8 incom 3303 . . . . . 6
97, 8eqtr3i 2278 . . . . 5
109uneq2i 3268 . . . 4
116, 10eqtr4i 2279 . . 3
125, 11eqtr4i 2279 . 2
13 ddif 3250 . . . . 5
1413uneq2i 3268 . . . 4
15 indm 3369 . . . . 5
16 invdif 3352 . . . . . 6
1716difeq2i 3233 . . . . 5
1815, 17eqtr3i 2278 . . . 4
1914, 18eqtr3i 2278 . . 3
2019ineq2i 3309 . 2
21 invdif 3352 . 2
2212, 20, 213eqtri 2280 1
 Colors of variables: wff set class Syntax hints:   wceq 1619  cvv 2740   cdif 3091   cun 3092   cin 3093  c0 3397 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398
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