Theorem difin0ss 1753

Description: Difference, intersection, and subclass relationship.

Assertion

Ref	Expression
difin0ss	|- (((A \ B) i^i C) = (/) -> (C (_ A -> C (_ B))

Proof of Theorem difin0ss

Step	Hyp	Ref	Expression
1		eq0 1719	. . 3 |- (((A \ B) i^i C) = (/) <-> A.x -. x e. ((A \ B) i^i C))
2		annim 206	. . . . . . . . 9 |- ((x e. A /\ -. x e. B) <-> -. (x e. A -> x e. B))
3	2	anbi2i 367	. . . . . . . 8 |- ((x e. C /\ (x e. A /\ -. x e. B)) <-> (x e. C /\ -. (x e. A -> x e. B)))
4		ancom 333	. . . . . . . 8 |- ((x e. C /\ (x e. A /\ -. x e. B)) <-> ((x e. A /\ -. x e. B) /\ x e. C))
5	3, 4	bitr3 153	. . . . . . 7 |- ((x e. C /\ -. (x e. A -> x e. B)) <-> ((x e. A /\ -. x e. B) /\ x e. C))
6	5	negbii 162	. . . . . 6 |- (-. (x e. C /\ -. (x e. A -> x e. B)) <-> -. ((x e. A /\ -. x e. B) /\ x e. C))
7		iman 205	. . . . . 6 |- ((x e. C -> (x e. A -> x e. B)) <-> -. (x e. C /\ -. (x e. A -> x e. B)))
8		elin 1635	. . . . . . . 8 |- (x e. ((A \ B) i^i C) <-> (x e. (A \ B) /\ x e. C))
9		eldif 1496	. . . . . . . . 9 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
10	9	anbi1i 368	. . . . . . . 8 |- ((x e. (A \ B) /\ x e. C) <-> ((x e. A /\ -. x e. B) /\ x e. C))
11	8, 10	bitr 151	. . . . . . 7 |- (x e. ((A \ B) i^i C) <-> ((x e. A /\ -. x e. B) /\ x e. C))
12	11	negbii 162	. . . . . 6 |- (-. x e. ((A \ B) i^i C) <-> -. ((x e. A /\ -. x e. B) /\ x e. C))
13	6, 7, 12	3bitr4 158	. . . . 5 |- ((x e. C -> (x e. A -> x e. B)) <-> -. x e. ((A \ B) i^i C))
14		ax-2 4	. . . . 5 |- ((x e. C -> (x e. A -> x e. B)) -> ((x e. C -> x e. A) -> (x e. C -> x e. B)))
15	13, 14	sylbir 176	. . . 4 |- (-. x e. ((A \ B) i^i C) -> ((x e. C -> x e. A) -> (x e. C -> x e. B)))
16	15	19.20ii 692	. . 3 |- (A.x -. x e. ((A \ B) i^i C) -> (A.x(x e. C -> x e. A) -> A.x(x e. C -> x e. B)))
17	1, 16	sylbi 174	. 2 |- (((A \ B) i^i C) = (/) -> (A.x(x e. C -> x e. A) -> A.x(x e. C -> x e. B)))
18		dfss2 1497	. 2 |- (C (_ A <-> A.x(x e. C -> x e. A))
19		dfss2 1497	. 2 |- (C (_ B <-> A.x(x e. C -> x e. B))
20	17, 18, 19	3imtr4g 426	1 |- (((A \ B) i^i C) = (/) -> (C (_ A -> C (_ B))

Syntax hints:  -. wn 1   -> wi 2   /\ 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:  tz7.7 2224  tfi 2244

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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