Theorem iindif2 2611

Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 2601 to recover Enderton's theorem.

Assertion

Ref	Expression
iindif2	|- (A =/= (/) -> |^|_x e. A (B \ C) = (B \ U_x e. A C))

Distinct variable groups:   x,A   x,B

Proof of Theorem iindif2

Step	Hyp	Ref	Expression
1		r19.28zv 2350	. . . 4 |- (A =/= (/) -> (A.x e. A (y e. B /\ -. y e. C) <-> (y e. B /\ A.x e. A -. y e. C)))
2		eldif 2057	. . . . 5 |- (y e. (B \ C) <-> (y e. B /\ -. y e. C))
3	2	ralbii 1667	. . . 4 |- (A.x e. A y e. (B \ C) <-> A.x e. A (y e. B /\ -. y e. C))
4		eliun 2570	. . . . . . 7 |- (y e. U_x e. A C <-> E.x e. A y e. C)
5	4	negbii 187	. . . . . 6 |- (-. y e. U_x e. A C <-> -. E.x e. A y e. C)
6		ralnex 1653	. . . . . 6 |- (A.x e. A -. y e. C <-> -. E.x e. A y e. C)
7	5, 6	bitr4 176	. . . . 5 |- (-. y e. U_x e. A C <-> A.x e. A -. y e. C)
8	7	anbi2i 480	. . . 4 |- ((y e. B /\ -. y e. U_x e. A C) <-> (y e. B /\ A.x e. A -. y e. C))
9	1, 3, 8	3bitr4g 555	. . 3 |- (A =/= (/) -> (A.x e. A y e. (B \ C) <-> (y e. B /\ -. y e. U_x e. A C)))
10		visset 1813	. . . 4 |- y e. V
11		eliin 2571	. . . 4 |- (y e. V -> (y e. |^|_x e. A (B \ C) <-> A.x e. A y e. (B \ C)))
12	10, 11	ax-mp 7	. . 3 |- (y e. |^|_x e. A (B \ C) <-> A.x e. A y e. (B \ C))
13		eldif 2057	. . 3 |- (y e. (B \ U_x e. A C) <-> (y e. B /\ -. y e. U_x e. A C))
14	9, 12, 13	3bitr4g 555	. 2 |- (A =/= (/) -> (y e. |^|_x e. A (B \ C) <-> y e. (B \ U_x e. A C)))
15	14	eqrdv 1473	1 |- (A =/= (/) -> |^|_x e. A (B \ C) = (B \ U_x e. A C))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  E.wrex 1646  Vcvv 1811   \ cdif 2044  (/)c0 2280  U_ciun 2566  |^|_ciin 2567

This theorem is referenced by:  iincld 7679

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-nul 2281  df-iun 2568  df-iin 2569

Copyright terms: Public domain