Theorem inssdif0 2333

Description: Intersection, subclass, and difference relationship.

Assertion

Ref	Expression
inssdif0	|- ((A i^i B) (_ C <-> (A i^i (B \ C)) = (/))

Proof of Theorem inssdif0

Step	Hyp	Ref	Expression
1		impexp 347	. . . . 5 |- (((x e. A /\ x e. B) -> x e. C) <-> (x e. A -> (x e. B -> x e. C)))
2		iman 237	. . . . . 6 |- ((x e. B -> x e. C) <-> -. (x e. B /\ -. x e. C))
3	2	imbi2i 185	. . . . 5 |- ((x e. A -> (x e. B -> x e. C)) <-> (x e. A -> -. (x e. B /\ -. x e. C)))
4		imnan 242	. . . . 5 |- ((x e. A -> -. (x e. B /\ -. x e. C)) <-> -. (x e. A /\ (x e. B /\ -. x e. C)))
5	1, 3, 4	3bitr 177	. . . 4 |- (((x e. A /\ x e. B) -> x e. C) <-> -. (x e. A /\ (x e. B /\ -. x e. C)))
6		elin 2207	. . . . 5 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
7	6	imbi1i 186	. . . 4 |- ((x e. (A i^i B) -> x e. C) <-> ((x e. A /\ x e. B) -> x e. C))
8		elin 2207	. . . . . 6 |- (x e. (A i^i (B \ C)) <-> (x e. A /\ x e. (B \ C)))
9		eldif 2057	. . . . . . 7 |- (x e. (B \ C) <-> (x e. B /\ -. x e. C))
10	9	anbi2i 480	. . . . . 6 |- ((x e. A /\ x e. (B \ C)) <-> (x e. A /\ (x e. B /\ -. x e. C)))
11	8, 10	bitr 173	. . . . 5 |- (x e. (A i^i (B \ C)) <-> (x e. A /\ (x e. B /\ -. x e. C)))
12	11	negbii 187	. . . 4 |- (-. x e. (A i^i (B \ C)) <-> -. (x e. A /\ (x e. B /\ -. x e. C)))
13	5, 7, 12	3bitr4 183	. . 3 |- ((x e. (A i^i B) -> x e. C) <-> -. x e. (A i^i (B \ C)))
14	13	albii 999	. 2 |- (A.x(x e. (A i^i B) -> x e. C) <-> A.x -. x e. (A i^i (B \ C)))
15		dfss2 2058	. 2 |- ((A i^i B) (_ C <-> A.x(x e. (A i^i B) -> x e. C))
16		eq0 2294	. 2 |- ((A i^i (B \ C)) = (/) <-> A.x -. x e. (A i^i (B \ C)))
17	14, 15, 16	3bitr4 183	1 |- ((A i^i B) (_ C <-> (A i^i (B \ C)) = (/))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958   \ cdif 2044   i^i cin 2046   (_ wss 2047  (/)c0 2280

This theorem is referenced by:  difdisj 2337  inf3lem3 4615  bcthlem9 8007

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-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281

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