Theorem difdifdir 2350

Description: Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16.

Assertion

Ref	Expression
difdifdir	|- ((A \ B) \ C) = ((A \ C) \ (B \ C))

Proof of Theorem difdifdir

Step	Hyp	Ref	Expression
1		difdisj 2341	. . . . 5 |- (C i^i (A \ C)) = (/)
2		incom 2211	. . . . 5 |- (C i^i (A \ C)) = ((A \ C) i^i C)
3	1, 2	eqtr3 1500	. . . 4 |- (/) = ((A \ C) i^i C)
4	3	uneq2i 2184	. . 3 |- (((A \ C) i^i (V \ B)) u. (/)) = (((A \ C) i^i (V \ B)) u. ((A \ C) i^i C))
5		invdif 2252	. . . 4 |- ((A \ C) i^i (V \ B)) = ((A \ C) \ B)
6		un0 2301	. . . 4 |- (((A \ C) i^i (V \ B)) u. (/)) = ((A \ C) i^i (V \ B))
7		dif23 2267	. . . 4 |- ((A \ B) \ C) = ((A \ C) \ B)
8	5, 6, 7	3eqtr4r 1509	. . 3 |- ((A \ B) \ C) = (((A \ C) i^i (V \ B)) u. (/))
9		indi 2254	. . 3 |- ((A \ C) i^i ((V \ B) u. C)) = (((A \ C) i^i (V \ B)) u. ((A \ C) i^i C))
10	4, 8, 9	3eqtr4 1508	. 2 |- ((A \ B) \ C) = ((A \ C) i^i ((V \ B) u. C))
11		indm 2265	. . . 4 |- (V \ (B i^i (V \ C))) = ((V \ B) u. (V \ (V \ C)))
12		invdif 2252	. . . . 5 |- (B i^i (V \ C)) = (B \ C)
13	12	difeq2i 2159	. . . 4 |- (V \ (B i^i (V \ C))) = (V \ (B \ C))
14		ddif 2172	. . . . 5 |- (V \ (V \ C)) = C
15	14	uneq2i 2184	. . . 4 |- ((V \ B) u. (V \ (V \ C))) = ((V \ B) u. C)
16	11, 13, 15	3eqtr3r 1507	. . 3 |- ((V \ B) u. C) = (V \ (B \ C))
17	16	ineq2i 2217	. 2 |- ((A \ C) i^i ((V \ B) u. C)) = ((A \ C) i^i (V \ (B \ C)))
18		invdif 2252	. 2 |- ((A \ C) i^i (V \ (B \ C))) = ((A \ C) \ (B \ C))
19	10, 17, 18	3eqtr 1502	1 |- ((A \ B) \ C) = ((A \ C) \ (B \ C))

Syntax hints:   = wceq 958  Vcvv 1814   \ cdif 2047   u. cun 2048   i^i cin 2049  (/)c0 2283

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284

Copyright terms: Public domain