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Theorem difdifdir 3541
 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 3431 . . . . 5
2 invdif 3410 . . . . 5
31, 2eqtr4i 2306 . . . 4
4 un0 3479 . . . 4
53, 4eqtr4i 2306 . . 3
6 indi 3415 . . . 4
7 disjdif 3526 . . . . . 6
8 incom 3361 . . . . . 6
97, 8eqtr3i 2305 . . . . 5
109uneq2i 3326 . . . 4
116, 10eqtr4i 2306 . . 3
125, 11eqtr4i 2306 . 2
13 ddif 3308 . . . . 5
1413uneq2i 3326 . . . 4
15 indm 3427 . . . . 5
16 invdif 3410 . . . . . 6
1716difeq2i 3291 . . . . 5
1815, 17eqtr3i 2305 . . . 4
1914, 18eqtr3i 2305 . . 3
2019ineq2i 3367 . 2
21 invdif 3410 . 2
2212, 20, 213eqtri 2307 1
 Colors of variables: wff set class Syntax hints:   wceq 1623  cvv 2788   cdif 3149   cun 3150   cin 3151  c0 3455 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456
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