Theorem ssdif0im 3308

Description: Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)

Assertion

Ref	Expression
ssdif0im	|- ( A C_ B -> ( A \ B ) = (/) )

Proof of Theorem ssdif0im

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imanim 818	. . . 4 |- ( ( x e. A -> x e. B ) -> -. ( x e. A /\ -. x e. B ) )
2		eldif 2982	. . . 4 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
3	1, 2	sylnibr 634	. . 3 |- ( ( x e. A -> x e. B ) -> -. x e. ( A \ B ) )
4	3	alimi 1384	. 2 |- ( A. x ( x e. A -> x e. B ) -> A. x -. x e. ( A \ B ) )
5		dfss2 2988	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
6		eq0 3266	. 2 |- ( ( A \ B ) = (/) <-> A. x -. x e. ( A \ B ) )
7	4, 5, 6	3imtr4i 199	1 |- ( A C_ B -> ( A \ B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   A.wal 1282   = wceq 1284   e. wcel 1433   \ cdif 2970   C_ wss 2973   (/)c0 3251

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-in 2979  df-ss 2986  df-nul 3252

This theorem is referenced by:  vdif0im  3309  difrab0eqim  3310  difid  3312  difin0  3317