Theorem difindiss 3218

Description: Distributive law for class difference. In classical logic, for example, theorem 40 of [Suppes] p. 29, this is an equality instead of subset. (Contributed by Jim Kingdon, 26-Jul-2018.)

Assertion

Ref	Expression
difindiss	|- ( ( A \ B ) u. ( A \ C ) ) C_ ( A \ ( B i^i C ) )

Proof of Theorem difindiss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 3113	. . 3 |- ( x e. ( ( A \ B ) u. ( A \ C ) ) <-> ( x e. ( A \ B ) \/ x e. ( A \ C ) ) )
2		eldif 2982	. . . . . . 7 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
3		eldif 2982	. . . . . . 7 |- ( x e. ( A \ C ) <-> ( x e. A /\ -. x e. C ) )
4	2, 3	orbi12i 713	. . . . . 6 |- ( ( x e. ( A \ B ) \/ x e. ( A \ C ) ) <-> ( ( x e. A /\ -. x e. B ) \/ ( x e. A /\ -. x e. C ) ) )
5		andi 764	. . . . . 6 |- ( ( x e. A /\ ( -. x e. B \/ -. x e. C ) ) <-> ( ( x e. A /\ -. x e. B ) \/ ( x e. A /\ -. x e. C ) ) )
6	4, 5	bitr4i 185	. . . . 5 |- ( ( x e. ( A \ B ) \/ x e. ( A \ C ) ) <-> ( x e. A /\ ( -. x e. B \/ -. x e. C ) ) )
7		pm3.14 702	. . . . . 6 |- ( ( -. x e. B \/ -. x e. C ) -> -. ( x e. B /\ x e. C ) )
8	7	anim2i 334	. . . . 5 |- ( ( x e. A /\ ( -. x e. B \/ -. x e. C ) ) -> ( x e. A /\ -. ( x e. B /\ x e. C ) ) )
9	6, 8	sylbi 119	. . . 4 |- ( ( x e. ( A \ B ) \/ x e. ( A \ C ) ) -> ( x e. A /\ -. ( x e. B /\ x e. C ) ) )
10		eldif 2982	. . . . 5 |- ( x e. ( A \ ( B i^i C ) ) <-> ( x e. A /\ -. x e. ( B i^i C ) ) )
11		elin 3155	. . . . . . 7 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
12	11	notbii 626	. . . . . 6 |- ( -. x e. ( B i^i C ) <-> -. ( x e. B /\ x e. C ) )
13	12	anbi2i 444	. . . . 5 |- ( ( x e. A /\ -. x e. ( B i^i C ) ) <-> ( x e. A /\ -. ( x e. B /\ x e. C ) ) )
14	10, 13	bitr2i 183	. . . 4 |- ( ( x e. A /\ -. ( x e. B /\ x e. C ) ) <-> x e. ( A \ ( B i^i C ) ) )
15	9, 14	sylib 120	. . 3 |- ( ( x e. ( A \ B ) \/ x e. ( A \ C ) ) -> x e. ( A \ ( B i^i C ) ) )
16	1, 15	sylbi 119	. 2 |- ( x e. ( ( A \ B ) u. ( A \ C ) ) -> x e. ( A \ ( B i^i C ) ) )
17	16	ssriv 3003	1 |- ( ( A \ B ) u. ( A \ C ) ) C_ ( A \ ( B i^i C ) )

Syntax hints:   -. wn 3   /\ wa 102   \/ wo 661   e. wcel 1433   \ cdif 2970   u. cun 2971   i^i cin 2972   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986

This theorem is referenced by:  difdif2ss  3221  indmss  3223