Theorem inssdif0 1754

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 276	. . . . 5 |- (((x e. A /\ x e. B) -> x e. C) <-> (x e. A -> (x e. B -> x e. C)))
2		iman 205	. . . . . 6 |- ((x e. B -> x e. C) <-> -. (x e. B /\ -. x e. C))
3	2	imbi2i 160	. . . . 5 |- ((x e. A -> (x e. B -> x e. C)) <-> (x e. A -> -. (x e. B /\ -. x e. C)))
4		imnan 207	. . . . 5 |- ((x e. A -> -. (x e. B /\ -. x e. C)) <-> -. (x e. A /\ (x e. B /\ -. x e. C)))
5	1, 3, 4	3bitr 155	. . . 4 |- (((x e. A /\ x e. B) -> x e. C) <-> -. (x e. A /\ (x e. B /\ -. x e. C)))
6		elin 1635	. . . . 5 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
7	6	imbi1i 161	. . . 4 |- ((x e. (A i^i B) -> x e. C) <-> ((x e. A /\ x e. B) -> x e. C))
8		elin 1635	. . . . . 6 |- (x e. (A i^i (B \ C)) <-> (x e. A /\ x e. (B \ C)))
9		eldif 1496	. . . . . . 7 |- (x e. (B \ C) <-> (x e. B /\ -. x e. C))
10	9	anbi2i 367	. . . . . 6 |- ((x e. A /\ x e. (B \ C)) <-> (x e. A /\ (x e. B /\ -. x e. C)))
11	8, 10	bitr 151	. . . . 5 |- (x e. (A i^i (B \ C)) <-> (x e. A /\ (x e. B /\ -. x e. C)))
12	11	negbii 162	. . . 4 |- (-. x e. (A i^i (B \ C)) <-> -. (x e. A /\ (x e. B /\ -. x e. C)))
13	5, 7, 12	3bitr4 158	. . 3 |- ((x e. (A i^i B) -> x e. C) <-> -. x e. (A i^i (B \ C)))
14	13	bial 695	. 2 |- (A.x(x e. (A i^i B) -> x e. C) <-> A.x -. x e. (A i^i (B \ C)))
15		dfss2 1497	. 2 |- ((A i^i B) (_ C <-> A.x(x e. (A i^i B) -> x e. C))
16		eq0 1719	. 2 |- ((A i^i (B \ C)) = (/) <-> A.x -. x e. (A i^i (B \ C)))
17	14, 15, 16	3bitr4 158	1 |- ((A i^i B) (_ C <-> (A i^i (B \ C)) = (/))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   = wceq 1091   e. wcel 1092   \ cdif 1484   i^i cin 1486   (_ wss 1487  (/)c0 1707

This theorem is referenced by:  difdisj 1758  inf3lem3 3466

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708

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