Theorem difindiss 3427

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 3314	. . 3 |- ( x e. ( ( A \ B ) u. ( A \ C ) ) <-> ( x e. ( A \ B ) \/ x e. ( A \ C ) ) )
2		eldif 3175	. . . . . . 7 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
3		eldif 3175	. . . . . . 7 |- ( x e. ( A \ C ) <-> ( x e. A /\ -. x e. C ) )
4	2, 3	orbi12i 766	. . . . . 6 |- ( ( x e. ( A \ B ) \/ x e. ( A \ C ) ) <-> ( ( x e. A /\ -. x e. B ) \/ ( x e. A /\ -. x e. C ) ) )
5		andi 820	. . . . . 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 187	. . . . 5 |- ( ( x e. ( A \ B ) \/ x e. ( A \ C ) ) <-> ( x e. A /\ ( -. x e. B \/ -. x e. C ) ) )
7		pm3.14 755	. . . . . 6 |- ( ( -. x e. B \/ -. x e. C ) -> -. ( x e. B /\ x e. C ) )
8	7	anim2i 342	. . . . 5 |- ( ( x e. A /\ ( -. x e. B \/ -. x e. C ) ) -> ( x e. A /\ -. ( x e. B /\ x e. C ) ) )
9	6, 8	sylbi 121	. . . 4 |- ( ( x e. ( A \ B ) \/ x e. ( A \ C ) ) -> ( x e. A /\ -. ( x e. B /\ x e. C ) ) )
10		eldif 3175	. . . . 5 |- ( x e. ( A \ ( B i^i C ) ) <-> ( x e. A /\ -. x e. ( B i^i C ) ) )
11		elin 3356	. . . . . . 7 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
12	11	notbii 670	. . . . . 6 |- ( -. x e. ( B i^i C ) <-> -. ( x e. B /\ x e. C ) )
13	12	anbi2i 457	. . . . 5 |- ( ( x e. A /\ -. x e. ( B i^i C ) ) <-> ( x e. A /\ -. ( x e. B /\ x e. C ) ) )
14	10, 13	bitr2i 185	. . . 4 |- ( ( x e. A /\ -. ( x e. B /\ x e. C ) ) <-> x e. ( A \ ( B i^i C ) ) )
15	9, 14	sylib 122	. . 3 |- ( ( x e. ( A \ B ) \/ x e. ( A \ C ) ) -> x e. ( A \ ( B i^i C ) ) )
16	1, 15	sylbi 121	. 2 |- ( x e. ( ( A \ B ) u. ( A \ C ) ) -> x e. ( A \ ( B i^i C ) ) )
17	16	ssriv 3197	1 |- ( ( A \ B ) u. ( A \ C ) ) C_ ( A \ ( B i^i C ) )

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 710   e. wcel 2176   \ cdif 3163   u. cun 3164   i^i cin 3165   C_ wss 3166

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179

This theorem is referenced by:  difdif2ss  3430  indmss  3432