Theorem difin 3400

Description: Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem difin

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-in2 616	. . . . . . . 8 |- ( -. ( x e. A /\ x e. B ) -> ( ( x e. A /\ x e. B ) -> F. ) )
2	1	expd 258	. . . . . . 7 |- ( -. ( x e. A /\ x e. B ) -> ( x e. A -> ( x e. B -> F. ) ) )
3		dfnot 1382	. . . . . . 7 |- ( -. x e. B <-> ( x e. B -> F. ) )
4	2, 3	imbitrrdi 162	. . . . . 6 |- ( -. ( x e. A /\ x e. B ) -> ( x e. A -> -. x e. B ) )
5	4	com12 30	. . . . 5 |- ( x e. A -> ( -. ( x e. A /\ x e. B ) -> -. x e. B ) )
6	5	imdistani 445	. . . 4 |- ( ( x e. A /\ -. ( x e. A /\ x e. B ) ) -> ( x e. A /\ -. x e. B ) )
7		simpr 110	. . . . . 6 |- ( ( x e. A /\ x e. B ) -> x e. B )
8	7	con3i 633	. . . . 5 |- ( -. x e. B -> -. ( x e. A /\ x e. B ) )
9	8	anim2i 342	. . . 4 |- ( ( x e. A /\ -. x e. B ) -> ( x e. A /\ -. ( x e. A /\ x e. B ) ) )
10	6, 9	impbii 126	. . 3 |- ( ( x e. A /\ -. ( x e. A /\ x e. B ) ) <-> ( x e. A /\ -. x e. B ) )
11		eldif 3166	. . . 4 |- ( x e. ( A \ ( A i^i B ) ) <-> ( x e. A /\ -. x e. ( A i^i B ) ) )
12		elin 3346	. . . . . 6 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
13	12	notbii 669	. . . . 5 |- ( -. x e. ( A i^i B ) <-> -. ( x e. A /\ x e. B ) )
14	13	anbi2i 457	. . . 4 |- ( ( x e. A /\ -. x e. ( A i^i B ) ) <-> ( x e. A /\ -. ( x e. A /\ x e. B ) ) )
15	11, 14	bitri 184	. . 3 |- ( x e. ( A \ ( A i^i B ) ) <-> ( x e. A /\ -. ( x e. A /\ x e. B ) ) )
16		eldif 3166	. . 3 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
17	10, 15, 16	3bitr4i 212	. 2 |- ( x e. ( A \ ( A i^i B ) ) <-> x e. ( A \ B ) )
18	17	eqriv 2193	1 |- ( A \ ( A i^i B ) ) = ( A \ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   F. wfal 1369   e. wcel 2167   \ cdif 3154   i^i cin 3156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163

This theorem is referenced by:  inssddif  3404  symdif1  3428  notrab  3440  disjdif2  3529  unfiin  6987  bj-charfundcALT  15455