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Theorem difdifdir 3828
 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 3679 . . . . 5
2 invdif 3657 . . . . 5
31, 2eqtr4i 2453 . . . 4
4 un0 3732 . . . 4
53, 4eqtr4i 2453 . . 3
6 indi 3662 . . . 4
7 disjdif 3812 . . . . . 6
8 incom 3598 . . . . . 6
97, 8eqtr3i 2452 . . . . 5
109uneq2i 3560 . . . 4
116, 10eqtr4i 2453 . . 3
125, 11eqtr4i 2453 . 2
13 ddif 3540 . . . . 5
1413uneq2i 3560 . . . 4
15 indm 3675 . . . . 5
16 invdif 3657 . . . . . 6
1716difeq2i 3523 . . . . 5
1815, 17eqtr3i 2452 . . . 4
1914, 18eqtr3i 2452 . . 3
2019ineq2i 3604 . 2
21 invdif 3657 . 2
2212, 20, 213eqtri 2454 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1437  cvv 3022   cdif 3376   cun 3377   cin 3378  c0 3704 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705 This theorem is referenced by: (None)
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