Theorem undif3 3515

Description: An equality involving class union and class difference. The first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 17-Apr-2012.)

Assertion

Ref	Expression
undif3	|- (A u. (B \ C)) = ((A u. B) \ (C \ A))

Proof of Theorem undif3

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 3220	. . . 4 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
2		pm4.53 478	. . . . 5 |- (-. (x e. C /\ -. x e. A) <-> (-. x e. C \/ x e. A))
3		eldif 3221	. . . . 5 |- (x e. (C \ A) <-> (x e. C /\ -. x e. A))
4	2, 3	xchnxbir 300	. . . 4 |- (-. x e. (C \ A) <-> (-. x e. C \/ x e. A))
5	1, 4	anbi12i 678	. . 3 |- ((x e. (A u. B) /\ -. x e. (C \ A)) <-> ((x e. A \/ x e. B) /\ (-. x e. C \/ x e. A)))
6		eldif 3221	. . 3 |- (x e. ((A u. B) \ (C \ A)) <-> (x e. (A u. B) /\ -. x e. (C \ A)))
7		elun 3220	. . . 4 |- (x e. (A u. (B \ C)) <-> (x e. A \/ x e. (B \ C)))
8		eldif 3221	. . . . 5 |- (x e. (B \ C) <-> (x e. B /\ -. x e. C))
9	8	orbi2i 505	. . . 4 |- ((x e. A \/ x e. (B \ C)) <-> (x e. A \/ (x e. B /\ -. x e. C)))
10		orc 374	. . . . . . 7 |- (x e. A -> (x e. A \/ x e. B))
11		olc 373	. . . . . . 7 |- (x e. A -> (-. x e. C \/ x e. A))
12	10, 11	jca 518	. . . . . 6 |- (x e. A -> ((x e. A \/ x e. B) /\ (-. x e. C \/ x e. A)))
13		olc 373	. . . . . . 7 |- (x e. B -> (x e. A \/ x e. B))
14		orc 374	. . . . . . 7 |- (-. x e. C -> (-. x e. C \/ x e. A))
15	13, 14	anim12i 549	. . . . . 6 |- ((x e. B /\ -. x e. C) -> ((x e. A \/ x e. B) /\ (-. x e. C \/ x e. A)))
16	12, 15	jaoi 368	. . . . 5 |- ((x e. A \/ (x e. B /\ -. x e. C)) -> ((x e. A \/ x e. B) /\ (-. x e. C \/ x e. A)))
17		simpl 443	. . . . . . 7 |- ((x e. A /\ -. x e. C) -> x e. A)
18	17	orcd 381	. . . . . 6 |- ((x e. A /\ -. x e. C) -> (x e. A \/ (x e. B /\ -. x e. C)))
19		olc 373	. . . . . 6 |- ((x e. B /\ -. x e. C) -> (x e. A \/ (x e. B /\ -. x e. C)))
20		orc 374	. . . . . . 7 |- (x e. A -> (x e. A \/ (x e. B /\ -. x e. C)))
21	20	adantr 451	. . . . . 6 |- ((x e. A /\ x e. A) -> (x e. A \/ (x e. B /\ -. x e. C)))
22	20	adantl 452	. . . . . 6 |- ((x e. B /\ x e. A) -> (x e. A \/ (x e. B /\ -. x e. C)))
23	18, 19, 21, 22	ccase 912	. . . . 5 |- (((x e. A \/ x e. B) /\ (-. x e. C \/ x e. A)) -> (x e. A \/ (x e. B /\ -. x e. C)))
24	16, 23	impbii 180	. . . 4 |- ((x e. A \/ (x e. B /\ -. x e. C)) <-> ((x e. A \/ x e. B) /\ (-. x e. C \/ x e. A)))
25	7, 9, 24	3bitri 262	. . 3 |- (x e. (A u. (B \ C)) <-> ((x e. A \/ x e. B) /\ (-. x e. C \/ x e. A)))
26	5, 6, 25	3bitr4ri 269	. 2 |- (x e. (A u. (B \ C)) <-> x e. ((A u. B) \ (C \ A)))
27	26	eqriv 2350	1 |- (A u. (B \ C)) = ((A u. B) \ (C \ A))

Syntax hints:  -. wn 3   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710   \ cdif 3206   u. cun 3207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215

This theorem is referenced by:  undifabs  3627