Theorem difin0ss 2939

Description: Difference, intersection, and subclass relationship.

Assertion

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

Proof of Theorem difin0ss

Step	Hyp	Ref	Expression
1		eq0 2889	. . 3 |- (((A \ B) i^i C) = (/) <-> A.x -. x e. ((A \ B) i^i C))
2		annim 257	. . . . . . . . 9 |- ((x e. A /\ -. x e. B) <-> -. (x e. A -> x e. B))
3	2	anbi2i 538	. . . . . . . 8 |- ((x e. C /\ (x e. A /\ -. x e. B)) <-> (x e. C /\ -. (x e. A -> x e. B)))
4		ancom 482	. . . . . . . 8 |- ((x e. C /\ (x e. A /\ -. x e. B)) <-> ((x e. A /\ -. x e. B) /\ x e. C))
5	3, 4	bitr3i 192	. . . . . . 7 |- ((x e. C /\ -. (x e. A -> x e. B)) <-> ((x e. A /\ -. x e. B) /\ x e. C))
6	5	notbii 204	. . . . . 6 |- (-. (x e. C /\ -. (x e. A -> x e. B)) <-> -. ((x e. A /\ -. x e. B) /\ x e. C))
7		iman 256	. . . . . 6 |- ((x e. C -> (x e. A -> x e. B)) <-> -. (x e. C /\ -. (x e. A -> x e. B)))
8		elin 2786	. . . . . . . 8 |- (x e. ((A \ B) i^i C) <-> (x e. (A \ B) /\ x e. C))
9		eldif 2609	. . . . . . . . 9 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
10	9	anbi1i 539	. . . . . . . 8 |- ((x e. (A \ B) /\ x e. C) <-> ((x e. A /\ -. x e. B) /\ x e. C))
11	8, 10	bitri 190	. . . . . . 7 |- (x e. ((A \ B) i^i C) <-> ((x e. A /\ -. x e. B) /\ x e. C))
12	11	notbii 204	. . . . . 6 |- (-. x e. ((A \ B) i^i C) <-> -. ((x e. A /\ -. x e. B) /\ x e. C))
13	6, 7, 12	3bitr4i 200	. . . . 5 |- ((x e. C -> (x e. A -> x e. B)) <-> -. x e. ((A \ B) i^i C))
14		ax-2 5	. . . . 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 218	. . . 4 |- (-. x e. ((A \ B) i^i C) -> ((x e. C -> x e. A) -> (x e. C -> x e. B)))
16	15	al2imi 1341	. . 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 216	. 2 |- (((A \ B) i^i C) = (/) -> (A.x(x e. C -> x e. A) -> A.x(x e. C -> x e. B)))
18		dfss2 2610	. 2 |- (C C_ A <-> A.x(x e. C -> x e. A))
19		dfss2 2610	. 2 |- (C C_ B <-> A.x(x e. C -> x e. B))
20	17, 18, 19	3imtr4g 612	1 |- (((A \ B) i^i C) = (/) -> (C C_ A -> C C_ B))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300   \ cdif 2590   i^i cin 2592   C_ wss 2593  (/)c0 2875

This theorem is referenced by:  tz7.7 3684  tfi 3937

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876

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