Theorem difin 2831

Description: Difference with intersection. Theorem 33 of [Suppes] p. 29. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
difin	|- (A \ (A i^i B)) = (A \ B)

Proof of Theorem difin

Step	Hyp	Ref	Expression
1		pm4.61 258	. . 3 |- (-. (x e. A -> x e. B) <-> (x e. A /\ -. x e. B))
2		anclb 356	. . . . 5 |- ((x e. A -> x e. B) <-> (x e. A -> (x e. A /\ x e. B)))
3		elin 2786	. . . . . 6 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
4	3	imbi2i 202	. . . . 5 |- ((x e. A -> x e. (A i^i B)) <-> (x e. A -> (x e. A /\ x e. B)))
5		iman 256	. . . . 5 |- ((x e. A -> x e. (A i^i B)) <-> -. (x e. A /\ -. x e. (A i^i B)))
6	2, 4, 5	3bitr2i 196	. . . 4 |- ((x e. A -> x e. B) <-> -. (x e. A /\ -. x e. (A i^i B)))
7	6	con2bii 238	. . 3 |- ((x e. A /\ -. x e. (A i^i B)) <-> -. (x e. A -> x e. B))
8		eldif 2609	. . 3 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
9	1, 7, 8	3bitr4i 200	. 2 |- ((x e. A /\ -. x e. (A i^i B)) <-> x e. (A \ B))
10	9	difeqri 2727	1 |- (A \ (A i^i B)) = (A \ B)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   \ cdif 2590   i^i cin 2592

This theorem is referenced by:  dfin4 2835  indif 2837  symdif1 2856  dfsdom2 5523

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-v 2294  df-dif 2597  df-in 2603

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