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Theorem difdifdir 3915
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 3760 . . . . 5
2 invdif 3738 . . . . 5
31, 2eqtr4i 2489 . . . 4
4 un0 3810 . . . 4
53, 4eqtr4i 2489 . . 3
6 indi 3743 . . . 4
7 disjdif 3900 . . . . . 6
8 incom 3690 . . . . . 6
97, 8eqtr3i 2488 . . . . 5
109uneq2i 3654 . . . 4
116, 10eqtr4i 2489 . . 3
125, 11eqtr4i 2489 . 2
13 ddif 3635 . . . . 5
1413uneq2i 3654 . . . 4
15 indm 3756 . . . . 5
16 invdif 3738 . . . . . 6
1716difeq2i 3618 . . . . 5
1815, 17eqtr3i 2488 . . . 4
1914, 18eqtr3i 2488 . . 3
2019ineq2i 3696 . 2
21 invdif 3738 . 2
2212, 20, 213eqtri 2490 1
Colors of variables: wff setvar class
Syntax hints:  =wceq 1395   cvv 3109  \cdif 3472  u.cun 3473  i^icin 3474   c0 3784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785
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