Theorem difindiss 3381

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 3268	. . 3 |- ( x e. ( ( A \ B ) u. ( A \ C ) ) <-> ( x e. ( A \ B ) \/ x e. ( A \ C ) ) )
2		eldif 3130	. . . . . . 7 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
3		eldif 3130	. . . . . . 7 |- ( x e. ( A \ C ) <-> ( x e. A /\ -. x e. C ) )
4	2, 3	orbi12i 759	. . . . . 6 |- ( ( x e. ( A \ B ) \/ x e. ( A \ C ) ) <-> ( ( x e. A /\ -. x e. B ) \/ ( x e. A /\ -. x e. C ) ) )
5		andi 813	. . . . . 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 186	. . . . 5 |- ( ( x e. ( A \ B ) \/ x e. ( A \ C ) ) <-> ( x e. A /\ ( -. x e. B \/ -. x e. C ) ) )
7		pm3.14 748	. . . . . 6 |- ( ( -. x e. B \/ -. x e. C ) -> -. ( x e. B /\ x e. C ) )
8	7	anim2i 340	. . . . 5 |- ( ( x e. A /\ ( -. x e. B \/ -. x e. C ) ) -> ( x e. A /\ -. ( x e. B /\ x e. C ) ) )
9	6, 8	sylbi 120	. . . 4 |- ( ( x e. ( A \ B ) \/ x e. ( A \ C ) ) -> ( x e. A /\ -. ( x e. B /\ x e. C ) ) )
10		eldif 3130	. . . . 5 |- ( x e. ( A \ ( B i^i C ) ) <-> ( x e. A /\ -. x e. ( B i^i C ) ) )
11		elin 3310	. . . . . . 7 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
12	11	notbii 663	. . . . . 6 |- ( -. x e. ( B i^i C ) <-> -. ( x e. B /\ x e. C ) )
13	12	anbi2i 454	. . . . 5 |- ( ( x e. A /\ -. x e. ( B i^i C ) ) <-> ( x e. A /\ -. ( x e. B /\ x e. C ) ) )
14	10, 13	bitr2i 184	. . . 4 |- ( ( x e. A /\ -. ( x e. B /\ x e. C ) ) <-> x e. ( A \ ( B i^i C ) ) )
15	9, 14	sylib 121	. . 3 |- ( ( x e. ( A \ B ) \/ x e. ( A \ C ) ) -> x e. ( A \ ( B i^i C ) ) )
16	1, 15	sylbi 120	. 2 |- ( x e. ( ( A \ B ) u. ( A \ C ) ) -> x e. ( A \ ( B i^i C ) ) )
17	16	ssriv 3151	1 |- ( ( A \ B ) u. ( A \ C ) ) C_ ( A \ ( B i^i C ) )

Syntax hints:   -. wn 3   /\ wa 103   \/ wo 703   e. wcel 2141   \ cdif 3118   u. cun 3119   i^i cin 3120   C_ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134

This theorem is referenced by:  difdif2ss  3384  indmss  3386