Theorem difin0ss 2329

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 2291	. . 3 |- (((A \ B) i^i C) = (/) <-> A.x -. x e. ((A \ B) i^i C))
2		annim 238	. . . . . . . . 9 |- ((x e. A /\ -. x e. B) <-> -. (x e. A -> x e. B))
3	2	anbi2i 480	. . . . . . . 8 |- ((x e. C /\ (x e. A /\ -. x e. B)) <-> (x e. C /\ -. (x e. A -> x e. B)))
4		ancom 435	. . . . . . . 8 |- ((x e. C /\ (x e. A /\ -. x e. B)) <-> ((x e. A /\ -. x e. B) /\ x e. C))
5	3, 4	bitr3 175	. . . . . . 7 |- ((x e. C /\ -. (x e. A -> x e. B)) <-> ((x e. A /\ -. x e. B) /\ x e. C))
6	5	negbii 187	. . . . . 6 |- (-. (x e. C /\ -. (x e. A -> x e. B)) <-> -. ((x e. A /\ -. x e. B) /\ x e. C))
7		iman 237	. . . . . 6 |- ((x e. C -> (x e. A -> x e. B)) <-> -. (x e. C /\ -. (x e. A -> x e. B)))
8		elin 2204	. . . . . . . 8 |- (x e. ((A \ B) i^i C) <-> (x e. (A \ B) /\ x e. C))
9		eldif 2054	. . . . . . . . 9 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
10	9	anbi1i 481	. . . . . . . 8 |- ((x e. (A \ B) /\ x e. C) <-> ((x e. A /\ -. x e. B) /\ x e. C))
11	8, 10	bitr 173	. . . . . . 7 |- (x e. ((A \ B) i^i C) <-> ((x e. A /\ -. x e. B) /\ x e. C))
12	11	negbii 187	. . . . . 6 |- (-. x e. ((A \ B) i^i C) <-> -. ((x e. A /\ -. x e. B) /\ x e. C))
13	6, 7, 12	3bitr4 183	. . . . 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 201	. . . 4 |- (-. x e. ((A \ B) i^i C) -> ((x e. C -> x e. A) -> (x e. C -> x e. B)))
16	15	19.20ii 994	. . 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 199	. 2 |- (((A \ B) i^i C) = (/) -> (A.x(x e. C -> x e. A) -> A.x(x e. C -> x e. B)))
18		dfss2 2055	. 2 |- (C (_ A <-> A.x(x e. C -> x e. A))
19		dfss2 2055	. 2 |- (C (_ B <-> A.x(x e. C -> x e. B))
20	17, 18, 19	3imtr4g 552	1 |- (((A \ B) i^i C) = (/) -> (C (_ A -> C (_ B))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957   \ cdif 2041   i^i cin 2043   (_ wss 2044  (/)c0 2277

This theorem is referenced by:  tz7.7 2969  tfi 3122

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-in 2048  df-ss 2050  df-nul 2278

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